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Water Expandable Polystyrene(WEPS)

Computational and ExperimentalAnalysis of Bubble Growth

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Snijders, Emile A.

Water expandable polystyrene (WEPS) : computational and experimentalanalysis of bubble growth / by Emile A. Snijders. - Eindhoven : TechnischeUniversiteit Eindhoven, 2003.Proefschrift. - ISBN 90-386-2934-6NUR 913

Trefwoorden: kunststoffen ; schuimvorming / blaasmiddelen ; water / bellen /fysisch-chemische simulatie en modellering / polystyreen / polyfenyleenether ;PPESubject headings: polymers ; foams / blowing agents ; water / bubbles /physicochemical simulation and modeling / polystyrene / poly(phenylene ether) ;PPE

c© 2003, E.A. SnijdersPrinted by Eindhoven University Press.

An electronic version of this thesis is available in PDF-format on the websiteof the Eindhoven University of Technology.

Water Expandable Polystyrene(WEPS)

Computational and ExperimentalAnalysis of Bubble Growth

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr. R.A. van Santen, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop dinsdag 15 april 2003 om 16.00 uur

door

Emile Antoine Snijders

geboren te Hengelo

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. P.J. Lemstraenprof.dr.ir. H.E.H. Meijer

Copromotor:dr. L.N.I.H. Nelissen

This research was financially supported by ’NOVA Chemicals’, Breda, The Netherlands.

Voor mijn moeder

Contents

Summary xi

1 Introduction 11.1 Polymer foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Expandable polystyrene (EPS) . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Methods for manufacturing EPS . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Preparation of expandable polystyrene beads . . . . . . . . . . . . . 21.3.2 Pre-expansion of polystyrene beads . . . . . . . . . . . . . . . . . . 31.3.3 Molding of pre-expanded beads . . . . . . . . . . . . . . . . . . . . 3

1.4 Blowing agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Water expandable polystyrene (WEPS) . . . . . . . . . . . . . . . . . . . . . 41.6 Modeling expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Survey of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Single bubble: Model description 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Schematic of a single bubble . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Maxwell constitutive equation and two limit situations . . . . . . . . 122.4.2 Multimode Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Extended Pom-Pom model . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Mass diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.1 Interfacial tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Single bubble growth: Asymptotical analysis 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Quadratic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.3 Hookean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Values for X and Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

viii Contents

3.4 Results: Asymptotical versus numerical time scales . . . . . . . . . . . . . . 303.4.1 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Hookean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.3 Maxwell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Error source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Single bubble growth: Numerical analysis 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Dimensionless equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Conservation and constitutive equations . . . . . . . . . . . . . . . . 424.2.2 Mass diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Dimensionless input parameters . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Computational techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4.2 Hookean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4.3 Maxwell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4.4 Multimode Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.5 Extended Pom-Pom model . . . . . . . . . . . . . . . . . . . . . . . 494.4.6 Including diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Results: Bubble growth for different constitutive models . . . . . . . . . . . 514.5.1 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5.2 Hookean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.5.3 Maxwell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.5.4 Maxwell model: From pure elastic to Newtonian . . . . . . . . . . . 574.5.5 Multimode Maxwell model . . . . . . . . . . . . . . . . . . . . . . 584.5.6 Extended Pom-Pom model . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Influence of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Suspension polymerization and expansion of WEPS 655.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1 Suspension polymerization . . . . . . . . . . . . . . . . . . . . . . . 655.1.2 Expansion of WEPS . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2.1 Suspension polymerization . . . . . . . . . . . . . . . . . . . . . . . 675.2.2 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3.1 Suspension polymerization . . . . . . . . . . . . . . . . . . . . . . . 715.3.2 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Modeling versus experiments 816.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.1.1 Computational analysis . . . . . . . . . . . . . . . . . . . . . . . . . 816.1.2 Experimental analysis . . . . . . . . . . . . . . . . . . . . . . . . . 816.1.3 Comparison of model and experiment . . . . . . . . . . . . . . . . . 82

6.2 Computations versus experiments . . . . . . . . . . . . . . . . . . . . . . . 82

Contents ix

6.2.1 EPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2.2 WEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Conclusions and recommendations . . . . . . . . . . . . . . . . . . . . . . . 86

A Mass diffusion 89A.0.1 Henry’s constant and diffusion coefficient . . . . . . . . . . . . . . . 91

B Asymptotical analysis considering the multimode Maxwell model 93

Bibliography 95

Technology assessment 99

Samenvatting 101

Dankwoord 103

Curriculum Vitae 105

x Contents

Summary

Styrofoam (polystyrene foam) is formed by the expansion of expandable polystyrene (EPS)beads, containing a volatile organic compound (VOC) as blowing agent, usually pentane.Upon heating the beads (e.g. with steam) above the glass transition temperature (Tg) andabove the boiling point of the blowing agent (Tbp), the beads expand. During this process,pentane is released into the environment contributing to the photochemical smog formationin the lower atmosphere. Since the mid-90’s, there is a strong tendency to refrain from usingVOC’s. In our laboratory, in collaboration with industry a new route has been developedobtaining water expandable polystyrene (WEPS), viz. polystyrene (PS) beads with water asthe sole blowing agent.Comparing EPS with WEPS reveals not only practical but also profound fundamental diffe-rences. A major difference is the solubility of the blowing agent in PS, water is immiscible,while pentane is miscible at elevated temperatures. For this reason, WEPS requires a discretedispersion of water in the PS matrix. Furthermore, pentane has a plasticizing effect on thePS matrix, reducing the glass transition temperature, while water has no such effect. Uponexpansion, the pentane content decreases, diminishing the plasticizing effect. Another im-portant difference is the gap (∼ 40K ) between the boiling point of the blowing agent and theglass transition temperature of the initial EPS bead, for WEPS no gap is present. For thesereasons, WEPS requires specific expansion conditions. To understand these differences inmore detail, the foaming of WEPS is modeled. The well-known single bubble model is usedas the framework for this analysis (asymptotical and numerical) concerning the growth of abubble in an infinite medium of a fluid (e.g. a polymer system).The asymptotical analysis, incorporating constitutive equations for the polymer matrix asgiven by a pure viscous (Newton), a pure elastic (Hooke) and a viscoelastic (Maxwell)model, results in the specific time scales for bubble growth. This analysis starts from anintegro-differential equation, which is a result of the momentum equation in combinationwith the continuity and the constitutive equations. Finding the essential time scales for bub-ble growth from this integro-differential equation in an analytical way requires an extensivemathematical method, which includes an approach to equilibrium, a Laplace transformationand an eigenfrequency problem respectively. This eigenfrequency problem results in a cubicequation, which is simplified to a set of analytically solvable equations by using a quadraticanalysis.The numerical analysis is performed to validate our asymptotical analysis (and vice versa) andto develop a framework that enables us to include heat and mass transfer and more complexconstitutive equations. The influences of different variables on the growth of the single bubblehave been investigated. From the numerical results, it is clear that diffusion has a majorimpact on the overall growth behavior of the single bubble, whereas other variables, such as

xii Summary

viscosity, modulus, external pressure and amount of blowing agent change the time scales forbubble growth and oscillation and change the maximum expandability.Moreover, our overall computational analysis is validated by experiments, by comparisonwith expansion of beads. The earlier developed WEPS-recipe is a two-step process. Inthe first step, water is emulsified by means of a suitable emulsifier in a pre-polymerizedstyrene/PS mixture possessing a high viscosity in order to fixate the emulsified water droplets.In the second step, this so-called inverse emulsion is suspended in water containing a suitablesuspension agent and the polymerization is completed. This recipe is improved for suspen-sion stability and reproducibility. Using this recipe, we prepared WEPS beads with varyingwater content and molar mass. The molar mass is related to the melt strength of the poly-mer matrix, changing the expansion behavior of the bead. Another possibility to influencethe melt strength is the dissolution of end-capped poly(2, 6-dimethyl-1, 4-phenylene ether)(PPE) in styrene prior to the pre-polymerization. This yields a homogeneous blend of PS andPPE containing water as the blowing agent, referred to as WE(PS/PPE), which is synthesizedsuccessfully for the first time.For all samples the expansion characteristics are determined by using our in-house developedlabscale expander (V ≈ 10ml). All beads have been expanded and recorded online using adigital camera. During the expansion process, three pronounced stages can be distinguished:(I) induction, (II) processing window and (III) collapse.Finally, the numerically calculated growth of the single bubble is qualitatively compared withthe experimentally observed expansion characteristics. Collapse is revealed in our numericalresults by including diffusion. Increasing the blowing agent concentration did experimentallynot result in a higher maximum expansion as computationally derived. This is expected to bea result of the more dominant effect of the water droplet size and dispersion. Experimentally,increasing the melt strength of the polymer system resulted in an increase of the processingwindow (Stage II) and a decreased collapse rate, which is in good agreement with our compu-tations. These comparisons and analogies have been derived from a relatively simple initialframework, and are therefore promising for continuing this research.In conclusion, despite the limited single bubble model, we are able, by comparing expe-riments and computations, to achieve valuable results concerning the expansion of WEPS.Moreover, the use of PPE of suitable molar mass to increase the melt strength might be ofvalue for the commercial PS/PPE foams, Dythermr, since this product also uses pentane asthe blowing agent.

Chapter 1

Introduction

1.1 Polymer foamsPolymer foams touch our lives every day. Some applications are hidden, such as the insulatingsheaths in houses and inside our refrigerators, while some applications are visible, such asprotective packaging and polystyrene hot cups. The plastic foam industry is a major segmentof the total plastic industry, accounting for about 10wt .% of the total plastic consumption.Polymer foams can be prepared from almost any polymer. The four most applied in the plasticindustry are polyurethane, polystyrene, polyolefin and polyvinyl chloride. In Figure 1.1 theU.S. market for polymer foam is presented (Forman, 2001).

PSfrag replacements

Polyolefin 4%Others 1%

Polyvinyl Chloride 16%

Polyurethane 53%

Polystyrene 26%

Figure 1.1: U.S. Market for polymer foam by plastic family, total is 3.37 million tonnes (for the year2001).

The polymer bulk density in polymer foams is decreased by the presence of numerous cellsdispersed throughout the polymer matrix. This can be achieved by using a variety of pro-cesses and so-called blowing agents, depending on the application. The main tool for allpreparations is the addition or generation of gas (the blowing agent) within the matrix. Theselection of both phases depends on their properties, their ease of manufacture and the eco-nomics of the total foaming system, whereas the polymer matrix may exist of more than onecomponent, as in the case of polymer blends or fillers (Benning, 1969).A wide spectrum of densities of the plastic foams are produced varying from about 960 downto 2kg/m3. This density in combination with the flexibility often determines the specific

2 Chapter 1

application limited by its mechanical strength. The cell geometry, cell size, shape and typeof cell are also important for the foam properties. Closed-cell foams, cells surrounded by athin layer of the solid polymer, are applied for e.g. thermal insulation, packaging and aregenerally rigid. Open-cell foams, the cells are connected via tunnels, are widely used in e.g.car seating, furniture, bedding and acoustic insulation and are generally flexible (Brody andMarsh, 1997).

1.2 Expandable polystyrene (EPS)Polystyrene (PS) was first commercially produced in the United States in 1938, about eightyears after its introduction in Germany. Earlier British patents issued in 1911 describe po-lystyrene resin, and literature references go back to the mid-nineteenth century. Polystyrenefoam was introduced in the early 1940s, and impact grades (PS enhanced with rubber to im-prove impact strength) have been commercialized in the United States after World War II(Pohleman and Echte, 1981).Expandable polystyrene (EPS) has captured a large market since their introduction nearly50 years ago. Nowadays about 2.5 million tonnes of raw material are processed world-wide(Gietl and Honl, 2001).Polystyrene foam is mostly manufactured from atactic polystyrene prepared via free radicalpolymerization in a suspension process. The molar mass and the molar mass distribution ofthe polystyrene matrix play a vital role in the properties of the foam. Since polystyrene is anorganic material it burns and since it is thermoplastic it melts.The excellent balance of cost and thermal insulation properties of polystyrene foams make itcompetitive with other commercial insulating materials, such as polyurethane foam and rockwool. The apolar properties of polystyrene in combination with a closed-cell structure makesit water resistant at ambient temperatures. For this reason it can be used for e.g. food pack-aging and floating applications. Since polystyrene foam has excellent shock-absorbing pro-perties, coupled with low costs and high insulation efficiency, it is ideal for low-temperatureinsulation.

1.3 Methods for manufacturing EPSThe two most important industrial processes for manufacturing polystyrene foams are mold-ing of pre-expanded beads and direct extrusion. Since we focus in this thesis on expandablebeads we will, from this point, continue with the description of the manufacturing process ofexpandable polystyrene beads (EPS) in more detail.

1.3.1 Preparation of expandable polystyrene beadsA batch-wise radical polymerization in a suspension process is a standard route for producingspherical expandable polystyrene beads. This route starts by using liquid styrene monomerdispersed in an aqueous medium containing a suitable suspension stabilizer, a hydrocarbonfoaming agent and a radical initiator. The mixture is heated to a suitable polymerizationtemperature. During the polymerization process a controlled hydrocarbon pressure is appliedto incorporate approximately 7wt .% of the blowing agent in the beads. A wide variety ofhydrocarbons have been applied as the blowing agent, e.g. butane, propane, propylene or

Introduction 3

more exotic ones like alcohols, esters and ketones. Frequently used are pentane isomerssince they possess the best cost/performance ratio. Nowadays a main serious disadvantage ofthis hydrocarbon (VOC: volatile organic compound) is its high inflammability and its volatilenature concerning the environment, since VOC’s contribute to the generation of ozone in thelower atmosphere (Babley, 1990).Manufacturers produce a wide range of expandable polystyrene grades, which differ in diam-eters and content of blowing agent, which determine the specific applications as illustrated inTable 1.1 (Frisch, 1972).

Table 1.1: Applications for EPS foam depending on the compact bead diameter.

Bead diameter [µm] Size classification Application800 − 2000 Large Insulation and building panels500 − 1000 Medium Molded items, e.g. packaging items300 − 800 Small Thin-walled containers, e.g. coffee cups

1.3.2 Pre-expansion of polystyrene beads

The density of the final foam is determined in the so-called pre-expansion stage. The pre-expansion of the compact beads in steam atmosphere is the most preferable way since steamtransfers, rapidly, relatively large quantities of energy to the beads. It is important that thebeads are heated rapidly to make sure that no blowing agent is lost, in order to generate foamwith a as low as possible density. Steam diffuses into the PS beads, ensuring a relativelyuniform temperature profile in the beads (Benning, 1969). In addition, the steam diffusedinto the beads increases the expandability of the beads (Crevecoeur, 1997).

1.3.3 Molding of pre-expanded beads

After the pre-expansion the beads contain approximately 4wt .% pentane and air at atmo-spheric pressure. Subsequently, the beads are matured, after which the pre-expanded beadsare entered into a mold. Steam is injected into the mold, allowing the beads to expand fur-ther and fuse together to fill the mold completely. The pressure of pentane and air are reducedwhile the steam pressure is maintained. After the foaming process has been completed, a con-trolled cooling of the system is of major importance since it determines the final performanceof the foam and the production capacity.

1.4 Blowing agents

Most thermoplastic polymer foams are prepared by adding a certain amount of blowing agentto the polymer. In general the blowing agents can be classified according to:

1. Chemical blowing agent: Generally a chemical compound that reacts at a certain tem-perature to produce a (inert) gas.

4 Chapter 1

2. Physical blowing agent: A gas or a low-boiling liquid is present in the polymer masseither by dissolving or by dispersion. At appropriate conditions this acts as the blowingagent.

Both types of blowing agent should at least meet the following requirements (Klempner andFrisch, 1991; Brody and Marsh, 1997):

• Long term storage stability.

• Gas release over controllable time and temperature range.

• No damaging effects on the stability and processing characteristics of the polymer.

• Economical feasibility.

• Environmental acceptability and safety.

The last mentioned requirement is one of the key features for the study presented in this thesis.With the ever-growing environmental regulations, the polymer industry already changed theblowing agents from chlorofluorocarbons (CFC’s) to volatile organic compounds (VOC’s)or mixtures of VOC’s and inert gases (Cohen, 2001). At this moment the polymer industrieshave to accept new regulations concerning VOC’s because these blowing agents are suspectedto be carcinogenic, to be flammable and to contribute to smog (Babley, 1990). To protectworkers and to minimize emissions, expensive plant ventilation and gas collection systemsare required.A more rigorous possibility is the replacement of the currently used (VOC) blowing agents.In many cases, existing processes involve more art than science. The highly soluble CFC’sand VOC’s are effective blowing agents, and their (partial) replacement by other blowingagents (e.g. C O2, H2O) is non-trivial. Water as a co-blowing agent in the manufacturingof polystyrene foam has been presented in literature (Wittenberg et al., 1992) reducing thenecessary content of the organic blowing agent. Although, an earlier patent (Keppler et al.,1977) describes that the incorporation of water has an adverse effect on the properties andfurther processing of the products.

1.5 Water expandable polystyrene (WEPS)Expandable polystyrene beads are manufactured by adding the blowing agent pentane dur-ing the suspension polymerization into the styrene/polystyrene mixture. This procedure forincorporating pentane will never work for water as the sole blowing agent since water andpolystyrene are immiscible. Therefore, water has to be introduced in the polystyrene matrixvia an alternative route. Crevecoeur et al. (1996) developed a process to incorporate waterin the early stages of the polymerization, stabilized by the addition of a suitable surfactant.This process is a trend in the EPS branche and several patents (Gluck et al., 1999a,b,c,d) referto this process. This yields spherical water droplets (micrometer scale) within the PS beads.The new polymerization procedure consists of the following main steps (see Figure 1.2):

1. Pre-polymerization in bulk of the styrene/emulsified water mixture to a conversion ofabout 30% in order to increase the viscosity of the matrix to stabilize the water droplets.

Introduction 5

2. Dispersion of the pre-polymerized mixture in water containing the appropriate suspen-sion stabilizer.

3. Free-radical suspension polymerization to complete the conversion of styrene, yieldingspherical polystyrene beads containing water as the unique blowing agent.

A

B

C

p r e - p o l y m e r i z a t i o n s u s p e n s i o n p o l y m e r i z a t i o n

A

B

C

p r e - p o l y m e r i z a t i o n s u s p e n s i o n p o l y m e r i z a t i o n

Figure 1.2: Schematic representation of the preparation process of WEPS, in the pre-polymerizationthe water (blowing agent) is emulsified in the styrene/PS mixture and in the suspensionpolymerization the PS beads containing emulsified water are formed, with A. Emulsifiedwater, B. Suspension medium (water), C. Organic styrene/PS phase.

It is obvious that the physical properties of water are different from the commercially usedhydrocarbons. Pentane is soluble in polystyrene at elevated temperatures, whereas water isnot. This requires a different dispersion of the blowing agent water throughout the polymermatrix. In the case of WEPS, the blowing agent is dispersed in the polymer matrix in spher-ical domains. EPS possesses a gap between the boiling point Tbp of the blowing agent andthe glass transition temperature Tg of the plasticized polymer. For water expandable poly-styrene (WEPS), the blowing agent has no plasticizing effect on polystyrene. In Table 1.2 acomparison is made for EPS and WEPS.

Table 1.2: The boiling point Tbp of the blowing agents and the glass transition temperatures Tg of theinitial polystyrene matrix for EPS and WEPS.

Tg [oC] Tbp [oC]EPS ∼ 70 28 − 36WEPS ∼ 100 ∼ 100

Upon expansion of the EPS beads the blowing agent pentane diffuses to the exterior of thebead. This causes an increase in Tg , since the plasticizing effect of pentane is concentrationdependent. This continuous up to the Tg of pure polystyrene, which is approximately equalto the temperature of the used heat source, steam, increasing the melt strength of the polymermatrix. The heating medium steam is not applicable for WEPS, since the temperature of thesteam is approximately equal to the Tg of the bead. In case of WEPS, the proposed heatingmedia are hot air and moreover superheated steam.

6 Chapter 1

1.6 Modeling expansion

Crevecoeur et al. (1999a) presented experimental results of the replacement in EPS of theconventional blowing agent pentane by water, resulting in water expandable polystyrene(WEPS). The replacement of this blowing agent changes the expansion behavior of polystyre-ne essentially. In order to gain more knowledge and to understand foaming of polystyrene ingeneral, a model is used as presented in this thesis.A large number of publications exists on the subject of modeling bubble growth and bubbleoscillation in different media. Street et al. (1971) established the initial framework for analyz-ing bubble growth. A finite envelope of fluid was placed around the bubble, later referred toas the single bubble model. They analyzed the coupled equation of motion, equation of conti-nuity and the equation of energy for this system. Since this early work, a lot of improvementshave been published. These publications included improved cell descriptions and more realis-tic conditions at the outer cell boundary (Amon and Denson, 1984), inclusion of viscoelasticeffects (Arefmanesh, 1991), analysis of bubble growth during process flow (Amon and Den-son, 1986; Arefmanesh et al., 1990), comparison of numerical foam expansion results withexperiments (Ramesh et al., 1991), simultaneous nucleation and bubble growth (Joshi et al.,1998) and complex rheological behavior (Agarwal, 2002).A typical polymer foaming process involves several steps. One of the first steps is the dis-solving of a gas blowing agent in the molten polymer after which respectively nucleation andbubble growth will occur upon a sudden release of pressure. For compact WEPS, the blow-ing agent (water is hardly soluble in polystyrene) is present in spherical domains inside thepolystyrene beads (Crevecoeur et al., 1999b) and thus no dissolving or nucleation has to beconsidered. Therefore, the expansion of compact WEPS starts when the temperature of thebead is above 100oC (the glass transition temperature of polystyrene and the boiling pointof the blowing agent water). At this moment the water present in the domains vaporizes,resulting in a driving force for expansion. This driving force decreases in strength due to thegrowth of the bubble and the diffusion of the blowing agent to the exterior the system. Theexpansion is partly resisted by the viscoelastic behavior of the polymer and the surface ten-sion. The growth of the bubble will continue until the forces resisting the expansion balancethe driving forces for expansion.

1.7 Objective

The objective of this thesis is to gain knowledge concerning the foaming of polymers. Inother words to predict the influences of different variables on the foaming, in order to be ableto control, whenever necessary, the existing systems in a more scientific way. This knowledgeis gained on the one hand by modeling and on the other hand by verifying the results of themodel with experimental results. WEPS is used as a model system.

1.8 Survey of thesis

As a starting point for modeling, the single bubble model has been chosen. In Chapter 2,this model is explained by using the conservation and constitutive equations. Also the inputparameters following from the chosen model system are presented.

Introduction 7

In Chapter 3, the asymptotical analysis is presented. This analysis uses the Maxwell consti-tutive model and two limiting situations, the Newtonian and Hookean model, to predict thetime scales for the growth of the single bubble in a simple, analytical way. This allows us toverify the numerical analysis as discussed in Chapter 4 and vice versa.In Chapter 4, the numerical analysis concerning the single bubble model is investigated. Dif-ferent numerical techniques are used to solve the growth of the single bubble using differentconstitutive equations, the Newtonian, Hookean, Maxwell and multimode Maxwell and ex-tended Pom-Pom model.In Chapter 5, different grades of water expandable polystyrene and derivatives are synthe-sized. A labscale expansion setup is used to foam the different grades and derivatives.Combining the results from Chapter 4 and the foaming results form Chapter 5 leads to acomparison between model and experiment. In the last chapter, Chapter 6, we predict andclarify the experimental outcome using the computational analysis and give a discussion onhow to improve our model system towards practical relevance.

8 Chapter 1

Chapter 2

Single bubble: Model description

2.1 Introduction

In this chapter, the initial framework for modeling is explained. At first, a schematic of thesingle bubble model is presented, after which the equations used are divided into the conser-vation laws, the constitutive equations, the mass diffusion equations, the input parameters forour model system (polystyrene for the shell and water as the core of the single bubble) andthe limitations of the framework presented.

2.2 Schematic of a single bubble

Foam in general consists of several bubbles in relatively close proximity to each other. Atthe start of the expansion, these bubbles can be considered as independent single sphericalbubbles in an infinite sea of incompressible fluid (e.g. polymer), till a certain moment in timeat which the bubbles start to deform. A schematic of the single bubble, is shown in Figure 2.1.

PSfrag replacements

pext

fluid

pg

gas

S � R(t)

R(t)

r

Figure 2.1: Schematic of the single bubble model, denoting the gas pressure pg , the external pressurepext , the radial component r , the bubble radius R(t) and the outer surface S.

Initially at time t = 0 the system is at rest, with (inner) bubble radius R0 and uniform externalpressure pext (equal to the constant atmospheric pressure).

10 Chapter 2

The description of the expansion of foam starts with a decrease in external pressure or anincrease in temperature. The bubble starts to expand, while gas will diffuse through the fluid.The expansion force is partly resisted by the (visco)elastic behavior of the fluid shell and thesurface tension effect. The growth of the bubble will continue until the forces resisting theexpansion balance the driving forces for expansion.We consider the gas pressure pg as function of the bubble radius R, with the ideal gas lawproviding this pressure

pg(R) =nB A RgT

43π R3

=αρB A R3

0MB A

RgTR3 =

κg

R3 , (2.1)

where ρB A is the density of the liquid blowing agent at STP (standard temperature and pres-sure, i.e. 273K and 105 Pa) in the initial bubble with radius R0, α is the initial fill fraction,which corresponds to the amount of blowing agent present in the initial bubble (100% fillequals α = 1), Rg is the universal gas constant, T is the constant temperature at which theexpansion takes place, MB A is the molar mass of the blowing agent and κg is introduced tosimplify Eq. 2.1.The fluid properties are assumed to be constant throughout the material and gravity can beneglected in comparison with the pressure gradient over the fluid.

2.3 Conservation lawsFor an incompressible fluid, the density ρ is constant, the equation of continuity reads

∇ · v = 0, (2.2)

in other words the divergence of v equals zero, where v is the velocity vector, containing thethree velocities described by the spherical coordinate system vr , vθ and vϕ . Irrespective ofthe detailed fluid rheology, the resulting radial velocity vr at radial position r in the fluid canbe obtained by integration of the equation of continuity, in the case of spherical symmetry

vr =R2 Rr2 , (2.3)

where the dot denotes the time derivative with R = d R/dt .The momentum equation, by neglecting the external forces such as the gravitational force, isgiven by

ρDv

Dt= −∇p + ∇ · τ , (2.4)

where the operator D/Dt is the substantial time derivative, ∇ · τ is the divergence of thedeviatoric stress for the fluid phase, ∇p is the gradient of the pressure and ρ is the fluiddensity. In the case of spherical symmetry and for an incompressible fluid Eq. 2.4 reduces to

ρ

(∂vr

∂ t+ vr

∂vr

∂r

)= −

∂p∂r

+ (∇ · τ)r . (2.5)

Single bubble: Model description 11

Integrating Eq. 2.5 for r between the bubble radius R and S, the system boundary (→ ∞),and substituting Eq. 2.3 leads to

32

R2 + RR =pl − pext

ρ+

1ρ

∫ ∞

R(∇ · τ )r dr, (2.6)

where the pressure at the fluid boundaries is given by p = pl at r = R and p = pext atr = S = ∞.By the usual force balance at the bubble-fluid interface, pl , the pressure in the fluid at r = R,can be obtained in terms of surface tension γ and the radial stresses, from

pl + τrr,l +2γ

R= pg(R) + τrr,g, (2.7)

where subscript g refers to the gas in the bubble and subscript l refers to the fluid phase. Sinceneither surface elasticity nor viscosity are considered in this analysis, the surface tension forceis given by the static surface tension γ .For a bubble containing ideal gas in a uniform state, the gas-phase stress at the bubble surfaceequals the pressure, thus τrr,g = 0 and the fluid-phase interfacial pressure is given by

pl = pg(R) −2γ

R− (τrr )r=R . (2.8)

Furthermore, the term (∇ · τ )r in Eq. 2.6 can be written in terms of the normal stresses as

(∇ · τ )r =∂τrr

∂r+

2τrr

r−

τθθ + τφφ

r. (2.9)

Now, upon substituting Eq. 2.8, 2.9 and 2.1 into Eq. 2.6 yields

32

R2 + RR =κg

R3ρ−

pext

ρ−

2γ

ρR+

1ρ

∫ ∞

R

(2τrr

r−

τθθ + τφφ

r

)dr. (2.10)

Eq. 2.10 is completely considered in the Chapters 3 and 4, although for highly viscous fluids,as polymer melts, we could neglect the inertia effects and the surface tension effects. Incor-porating both terms in our analysis, makes it in general valid for a wide range of materials.To complete the description of motion, we need to relate the fluid-phase stresses to the bubblemotion using a proper constitutive equation in combination with initial conditions.

2.4 Constitutive equationsAn analysis involving (visco)elastic fluids requires a relation between the stress and the his-tory of the motion. We use the usual material coordinates, where r(t ′) denotes the positionat a past time t ′ and its position r at the present time t (0 ≤ t ′ ≤ t). For a velocity field asdescribed by Eq. 2.3, assuming an incompressible fluid (conservation of volume and mass),one finds in spherical coordinates

r3(t ′) = r 3 + R3(t ′) − R3. (2.11)

This equation is visualized in Figure 2.2.

12 Chapter 2

PSfrag replacements

t ′ t

R rR r

Figure 2.2: In time the bubble expands from R(t ′) to R(t) (t ′ < t) while the material follows asindicated by material coordinate r(t ′) which moves to r(t)

The equi-biaxial rate of deformation tensor D is given by

D =12(L + LT ) =

ε 0 00 − 1

2 ε 00 0 − 1

2 ε

, (2.12)

in which L is the velocity gradient tensor and (·)T denotes the transpose of a tensor. For ourcase: L = LT = D and at any position (r, t) the rate of strain is given by,

ε(r, t) =∂vr

∂r= −2

RR2

r3 . (2.13)

and the history of the rate of strain is, given Eq. 2.11 and Eq. 2.13, determined by

ε(r ′, t ′) = −2R(t ′)R2(t ′)

r3 + R3(t ′) − R3 . (2.14)

The stress tensor τ is given by

τ =

τrr 0 00 τθθ 00 0 τφφ

, (2.15)

where due to symmetry of bubble growth τθθ equals τφφ . For the special case of sphericalsymmetry, a linear constitutive equation and an incompressible fluid, the sum of the deviatoricstresses is by definition equal to zero, thus one can express the φ and θ stresses in the termsof the radial stress as

−12τrr = τθθ = τφφ . (2.16)

2.4.1 Maxwell constitutive equation and two limit situationsConsidering a simple linear viscoelastic fluid model, we can write the normal radial stressrelated to the corresponding rate of strain using an integral form by

τrr (r, t) = 2∫ t

0N(t − t ′)ε(r ′, t ′)dt ′, (2.17)


where N(t) is a memory function or relaxation modulus. To describe Maxwell materialbehavior, this memory function is defined as follows

N(t) = G0e−t/λ0, (2.18)

where λ0 denotes the relaxation time and G0 the plateau modulus. Remember: the relaxationtime λ0 is equal to the zero-shear viscosity η0 divided by the plateau modulus G0 for a simpleviscoelastic material.Another way of writing the normal radial stress related to the corresponding rate of strain forMaxwell material behavior is the differential form

λ0dτ

dt+ τ = 2η0 D, (2.19)

which can be visualized using a spring and a dashpot as can be seen in Figure 2.3.

Newtonian model Hookean model Maxwell model multimode Maxwell model

PSfrag replacements

1

2

N

Figure 2.3: The spring (G0) and dashpot (η0) representation of the Newtonian, Hookean, Maxwell andmultimode Maxwell model respectively, where N denotes the total number of modes.

The multimode Maxwell model is represented by several Maxwell elements parallel.Now, using the integral form of the constitutive equation for Maxwell material behavior(Eq. 2.17 and 2.18) in combination with Eq. 2.14 yields

τrr (r, t) = −4G0

∫ t

0e−(t−t ′)/λ0

R(t ′)R2(t ′)r3 + R3(t ′) − R3 dt ′, (2.20)

and the integral in Eq. 2.10 in combination with Eq. 2.16 becomes

∫ ∞

R

τrr

rdr = −4G0

∫ t

0e−(t−t ′)/λ0

R(t ′)R2(t ′) ln[R(t ′)/R

]

R3(t ′) − R3 dt ′. (2.21)

Under these conditions the complete equation governing the expansion of the single bubblein a large body of an incompressible fluid (given polymer), acting as a Maxwell material, isgiven by

32

R2 + RR =κg

ρR3 −pext

ρ−

2γ

ρR

−12G0

ρ

∫ t

0e−(t−t ′)/λ0

R(t ′)R2(t ′) ln[R(t ′)/R

]

R3(t ′) − R3 dt ′.

(2.22)

14 Chapter 2

In view of the number of parameters we will also consider two limit situations, purely viscous(Newtonian λ0 → 0) and purely elastic material behavior (Hookean λ0 → ∞).Fluids with a very short relaxation time behave essentially as a purely viscous material, thiscorresponds to λ0 → 0. We substitute λ0 = η0/G0 into Eq. 2.22, this leads to

32

R2 + RR =κg

ρR3 −pext

ρ−

2γ

ρR

−12η0

ρ

∫ t

0

e−(t−t ′)/λ0

λ0

R(t ′)R2(t ′) ln[R(t ′)/R

]

R3(t ′) − R3 dt ′.

(2.23)

Now, considering λ0 → 0, we introduce the following identity

limλ0→0

e(−t+t ′)/λ0

λ0= δ(t − t ′), (2.24)

where δ denotes the Dirac delta function. Thus, Eq. 2.23 for short relaxation times becomes

32

R2 + RR =κg(R)

ρR3 −pext

ρ−

2γ

ρR−

4η0 RρR

. (2.25)

Fluids with a very long relaxation time behave essentially as a purely elastic material, thiscorresponds to λ0 → ∞ in Eq. 2.22, the equation of motion becomes

32

R2 + RR =κg

ρR3 −pext

ρ−

2γ

ρR

−12G0

ρ

∫ t

0

R(t ′)R2(t ′) ln[R(t ′)/R

]

R3(t ′) − R3 dt ′.

(2.26)

2.4.2 Multimode Maxwell

The multimode approximation is often necessary for a realistic description of the viscoelasticcontributions. The correlation between the fluid-phase stresses and the bubble motion for amultimode Maxwell model is given by

λ0,idτ i

dt+ τ i = 2η0,i D, (2.27)

where the viscoelastic contribution of the i th relaxation mode is denoted by τ i . The stressesin the fluid can now be described as

τ =N∑

i=1

τ i , (2.28)

where N denotes the total number of different modes present.


2.4.3 Extended Pom-Pom modelThe growth of the single bubble is described by biaxial extension. For this reason, we needa constitutive equation for the rheology of the polymer melt, that correctly describes thenon-linear behavior in both elongation and shear. Recently, McLeish and Larson (1998)introduced a new constitutive model, which is a major step forward in solving this problem:the Pom-Pom model. (Verbeeten et al., 2001) improved the original differential equationsresulting in the extended Pom-Pom (XPP) model. Although, the Pom-Pom model has beendeveloped for branched polymers, it is quite capable of predicting experimental data of linearpolymers (e.g. high density polyethylene). XPP uses also a multimode approximation ofthe relaxation spectrum as shown for the multimode Maxwell model. Below, the constitutivebehavior for a single mode of the viscoelastic contribution is described with the XPP model

∇τ +λ(τ)−1 · τ = 2G0 D, (2.29)

where∇τ is the upper convected time derivative of τ and is given by

∇τ= τ − L · τ − τ · LT =

dτ

dt− L · τ − τ · LT , (2.30)

where for a Lagrangian discretization we can define dτ/dt = ∂τ/∂ t and the dot denotes thescalar product. The relaxation tensor λ(τ)−1 is defined as:

λ(τ)−1 =1

λ0b

[α

G0τ + f (τ )−1 I + G0

(f (τ)−1 − 1

)τ

−1]

, (2.31)

where λ0b is the relaxation time of the backbone tube orientation, α is a material parameter(α ≥ 0), defining the amount of anisotropy and I is the unit tensor.The extra function f (τ ) present in Eq. 2.31 is defined as

1λ0b

f (τ)−1 =2λs

(1 −

13

)+

1λ0b32

[1 −

α Iτ ·τ

3G20

], (2.32)

where Iτ ·τ is the first invariant of (τ · τ ), equal to the trace of (τ · τ) and given by

tr(τ · τ ) = τ 2rr + τ 2

θθ + τ 2φφ = τ 2

rr + 2τ 2θθ . (2.33)

and the backbone stretch 3 and stretch relaxation time λs are given by

3 =√

1 + Iτ3G0

,

λs = λ0se−ν(3−1),

(2.34)

where Iτ is the first invariant of τ , equal to the trace of τ which is given by,

trτ = τrr + τθθ + τφφ = τrr + 2τθθ , (2.35)

and

ν =2q

, (2.36)

where q is the number of arms at the end of a backbone. Alternatively, ν can be taken as ameasure of the influence of the surrounding polymer chains on the backbone tube stretch.

16 Chapter 2

2.5 Mass diffusionTo determine the decrease in amount of gas present in the bubble, the principle of conserva-tion of mass is applied to the gas inside the bubble. The concentration gradient at the cellboundary can be related to the change in gas amount in the bubble (Bird et al., 1960)

ddt

(ρg R3

)= 3ρB A DR2 ∂ci(r, t)

∂r|r=R, (2.37)

where ρg is the density of the gas inside the bubble, ρB A is the density of the blowing agentat STP, D is the diffusion coefficient of the gas through the polymer, and ∂ci(r, t)/∂r |r=R isthe concentration gradient of the gas at the bubble interface. The left-hand-side of Eq. 2.37 isthe rate of accumulation of mass inside the bubble (which is negative due to loss of blowingagent to the surroundings) and the right-hand-side is the rate of diffusion of gas from theinside of the bubble to the surrounding polymer matrix. The diffusion of the gas through thepolymer matrix is governed by the following equation in spherical coordinates

∂ci(r, t)∂ t

+ vr∂ci(r, t)

∂r=

Dr2

∂

∂r

(r2 ∂ci(r, t)

∂r

). (2.38)

There are different approaches for solving the diffusion equation, in what follows, the ap-proximate method of solution will be adopted (Han and Yoo, 1981) resulting in

ddt

(ρg R3

)=

6ρB A DR2(c0 − cw)

δ, (2.39)

where δ is the thin concentration boundary layer and is approximated in Appendix A.

2.6 Input parametersAs material parameters we chose the external pressure equal to the atmospheric pressure,the density of polystyrene equal to 1000kg/m3 and independent of the temperature. For theliquid blowing agent we chose the density and the molar mass equal to water. Concerningthe material behavior for polystyrene, we considered 172N supplied by NOVA Chemicals(Mw = 260kg/mol).The shear rheology can be measured at various temperatures using small amplitude oscilla-tory shear devices. For the polystyrene example (Figure 2.4), a Rheometrics device equippedwith 25mm parallel plates was used at 390, 410, 430, 450, 470K . All polystyrene test sam-ples were prepared via compression molding of 25mm wide by 2mm thick disks. At eachtemperature the dynamic shear viscosity as a function of the shear frequency was measured.The data sets are reduced to one master curve at 410K , the reference temperature.Amorphous materials, like polystyrene fulfill the empirical Cox-Merz rule (Ferry, 1980),stating that the magnitude of the dynamic viscosity η∗ is equal to the steady viscosity η atcorresponding values of frequency and shear rate

η∗(ω) = η(γ ), (2.40)

with ω = γ .


10−4

10−2

100

102

104

102

103

104

105

106

107

108

PSfrag replacements

frequency, ω [s−1]

◦η

∗[P

a·s

],×

G′ [

Pa]

,+G

′′[P

a]

Figure 2.4: Shear viscosity (η∗, ◦) and loss (G ′′, +) and storage (G ′, ×) modulus for polystyrene 172Nat 410K as obtained by WLF shifting from Rheometrics 25mm parallel plate data at 390,410, 430, 450, 470K .

The zero-shear viscosity can be determined from the provided data by fitting η∗ using theCross model

η∗ − η∞η0 − η∞

=1

1 + (Kω)(1−n), (2.41)

where η∞ is the viscosity at infinite shear rate, K is a characteristic time constant and n isthe power law index. Fitting the experimental data according to Eq. 2.41 results in the valuesgiven in Table 2.1.

Table 2.1: Parameters for fitting the shear-viscosity-shear rate master curve at 410K using the Crossmodel.

η0 [Pa · s] 1.78 · 107

η∞ [Pa · s] 1.23 · 103

K [s] 5.59 · 102

n [−] 2.53 · 10−1

The effect of temperature on the plateau modulus (G0) is determined by the min-tan(δ)

method for the different experimental temperatures (Eckstein et al., 1998).The effect of temperature on viscoelastic material functions is modelled by the free-volumeavailability, and defined by Williams-Landel-Ferry (WLF) equation (Ferry, 1980), which isgiven by

log aT =−c0

1(T − T0)

c02 + T − T0

. (2.42)

The experimentally determined c01 and c0

2 are 13.87 and 52.21K , respectively. These valueslie within the range reported by Ferry (1980).The zero-shear viscosity and the plateau modulus are summarized in Table 2.2.

18 Chapter 2

Table 2.2: The plateau modulus (G0) and the zero-shear viscosity (η0) for different temperatures.

T [K ] G0[N/m2] η0[Pa · s]400 2.82 · 105 3.44 · 1010

405 2.80 · 105 5.24 · 108

410 2.77 · 105 1.78 · 107

415 2.74 · 105 1.09 · 106

420 2.72 · 105 1.05 · 105

425 2.70 · 105 1.43 · 104

The shear viscosity-shear rate master curve can be fitted to a multi-mode Maxwell model.For this model, the dynamic moduli are given by

G′ =N∑

i=1

G0,iλ2

i ω2

1 + λ2i ω

2, (2.43)

G′′ =N∑

i=1

G0,iλiω

1 + λ2i ω

2. (2.44)

The complex viscosity is defined as

η∗ = [(G′′

ω)2 + (

G′

ω)2]1/2 =

G∗

ω, (2.45)

with G∗ the complex modulus.Fitting the experimental data using a least square method according to Equation 2.43 and2.44 yields the Maxwell parameters (G0,i , λ0,i ), see Table 2.3. These individual discreterelaxation times have no physical meaning. The combination of all modes represents thecontinuous relaxation behavior of the material (Swartjes, 2001).

2.6.1 Interfacial tensionFurthermore, we need the interfacial tension γ for the polystyrene-water interface as functionof temperature [K ]. The interfacial tension (Wu, 1970) in m N/m is given by

γ = 40.7 − 0.072(T − 293). (2.46)

2.6.2 DiffusionWe incorporate diffusion of our blowing agent to the surrounding of the single bubble in ournumerical analysis. The diffusion of the blowing agent is independent of the constitutivedescription of our polymer shell. For this reason we only considered the momentum equationincorporating the Newton model in combination with mass diffusion.An important property for the gas polymer system including diffusion is the Henry’s lawconstant (kh). Solubilities of gases in polymer systems depend strongly on the temperature


Table 2.3: Parameters of the fits for the multimode Maxwell model for different number of modes N atT = 410K .

N G0,i[Pa] λ0,i [s]1 2.75 · 104 2.66 · 101

1 3.45 · 106 2.06 · 10−4

2 4.12 · 103 6.68 · 102

1 4.79 · 106 1.04 · 10−4

2 9.10 · 104 5.12 · 10−2

3 2.41 · 104 1.70 · 101

4 3.82 · 103 5.34 · 102

1 5.03 · 106 8.54 · 10−5

2 1.47 · 105 1.11 · 10−2

3 2.36 · 104 4.73 · 10−1

4 1.72 · 104 1.01 · 101

5 8.50 · 103 1.06 · 102

6 1.50 · 103 9.65 · 102

of the system. The experimental data as found in literature by Pogany (1976) and the fit asproposed by Krevelen van (1990) match as can be seen in Appendix A.Another important property for the gas polymer system including diffusion is the diffusioncoefficient (D). Experimental data as found in literature by Pogany (1976) is extrapolatedusing the free volume theory as proposed by Krevelen van (1990) and is presented in Ap-pendix A.We have to be careful by using the proposed fits, since the free volume theory cannot describethe diffusion involving small molecules, such as water, in amorphous polymers (Vrentas andDuda, 1976; Sok, 1994). The minimum void volume necessary for movement of very smallpenetrant molecules is less than the average void volume of the system (Vrentas and Duda,1976; Frisch et al., 1971).In our numerical analysis (Chapter 4) we will use the proposed data only as starting points(at T around 420K : kh ≈ 10−7cm3/(cm3 · Pa) and D ≈ 10−9m2/s), from which we startto investigate the influence of permeability (equal to solubility times diffusivity).

2.7 Limitations

We consider a single bubble in an infinite mass of polymer fluid. This is perhaps not theframework expected for modeling WEPS, which consists of several bubbles (the domainscontaining the blowing agent, water) in close proximity, influencing each other upon expan-sion. For this reason, we have to consider the conclusions deriving from this work with greatcare. However, from our results we wil be able to predict the influences of several parametersin a qualitative way.At the same time we do not neglect parts of the equations used, which from the model systempoint of view could be neglected, e.g. the inertia and surface tension effects, which are likely

20 Chapter 2

to be of very small influence for highly viscous fluids. This generalizes our framework tobubble expansion in arbitrary media.

Chapter 3

Single bubble growth:Asymptotical analysis

3.1 IntroductionThe momentum equation including Maxwell material behavior as given in the previous chap-ter (Eq. 2.22) is an integro-differential equation which can be solved numerically, see Chap-ter 4. To solve this equation in an analytical way we use a series of assumptions and mathe-matical techniques.To guide you through this chapter, a flow scheme of the used assumptions and mathematicaltechniques is shown:

1. First we assumed the lay-out of the single bubble as presented in Chapter 2, applyingspherical symmetry. Further, we assumed that the blowing agent can be described usingthe ideal gas law, and the polymer matrix using the Maxwell model. This resulted inEq. 2.22, which is an integro-differential equation.

2. We will start using an approach to equilibrium. This results in a linear integro-differential equation, depending on the deviation from the equilibrium radius u, asgiven by Eq. 3.10.

3. Now, performing a Laplace transformation simplifies Eq. 3.10 to a eigenfrequencyproblem.

4. This eigenfrequency problem can be rewritten into a cubic equation (Eq. 3.21), de-pendent on two dimensionless variables X and Y (see Eq. 3.20).

5. From this equation, we can consider the limits for X and Y by applying a quadraticanalysis (3.2.1) resulting in analytically soluble quadratic equations as summarized inTable 3.1.

We will verify our analysis using not only the limiting cases of the Maxwell model, theNewtonian and Hookean model, but also using the result of a full numerical analysis whosedetails are discussed in Chapter 4.

22 Chapter 3

3.2 Theoretical analysisWe start with the integro-differential equation (Eq. 2.22)

32

R2 + RR =κg

ρR3 −pext

ρ−

2γ

ρR

−12G0

ρ

∫ t

0e−(t−t ′)/λ0

R(t ′)R2(t ′) ln[R(t ′)/R

]

R3(t ′) − R3 dt ′,

(3.1)

which combines the conservation laws and the constitutive equation for the Maxwell model.In this equation we can discover an external part, given by

ϕ(R) =κg

R3 − pext −2γ

R. (3.2)

We now introduce the equilibrium radius, i.e. the radius to which the bubble tends wheninertia effects can be neglected and which is reached asymptotically for t → ∞. In theNewtonian and Maxwell case this equilibrium radius Req depends only on the external partas given in Eq. 3.2 and is found as the square root of ϕ(R) = 0 which is a cubic equation in1/R.In the Hookean case, as defined in Eq. 2.26, the external part has an extra term as defined inEq. 3.3

ϕH (R) =κg

R3 − pext −2γ

R+ 4G0

∫ ∞

Rln

[r3 − R3 + R3

0r3

]drr

, (3.3)

which, consequently, has a different equilibrium radius.Given the inherent complexity of the momentum equation incorporating the constitutiveMaxwell equation (Eq. 3.1), we have to focus on the approach to equilibrium by adoptinga so-called asymptotical analysis. Our starting point is

R(t) = Req + u(t), (3.4)

where |u(t)| � Req is the difference from equilibrium. Substituting Equation 3.4 into themomentum equation using Maxwell material behavior (Eq. 3.1) results in

ρ

(32

u2(t) + Req u(t) + u(t)u(t))

= ϕ(Req + u(t)

)

−12G0

∫ t

0e(−t+t ′)/λ0

u(t ′)(Req + u(t ′))2 ln[

Req+u(t ′)Req+u(t)

]

(Req + u(t ′)

)3 −(Req + u(t)

)3 dt ′.

(3.5)

Neglecting all contributions of u(t) higher than linear order, and using the expansion

ln[

Req + u(t ′)Req + u(t)

]= ln

[1 +

u(t ′) − u(t)Req

]≈

u(t ′) − u(t)Req

, (3.6)

Single bubble growth: Asymptotical analysis 23

since |u(t)| ,∣∣u(t ′)

∣∣ � Req , we find

ρReq u(t) = ϕ(Req + u(t)) −4G0

Req

∫ t

0e(t ′−t)/λ0 u(t ′)dt ′. (3.7)

To find a linear contribution for ϕ(Req + u(t)) we use

ϕ(Req + u(t)) ≈ ϕ(Req) + ϕ′(Req)u(t) + ... (3.8)

with ϕ(Req) = 0 for the Maxwell and Newton model and

ϕ′(Req) =∂ϕ(Req)

∂ Req= −

3κg

R4eq

+2γ

R2eq

< 0. (3.9)

Eq. 3.8 is expected to be a very good approximation for systems where fluctuations aroundequilibrium (Req ) are small (→ 0) for asymptotical time scales (Newtonian and Maxwellmaterial behavior). This approximation should be considered more carefully for Hookeanmaterial behavior, since we expect non-harmonic and undamped oscillations around the equi-librium radius. In real life (e.g. polystyrene or soap bubble) we will always observe dampingof a given oscillation. Therefore, we continue using Eq. 3.8 and will check this approxima-tion with particular care for the Hookean case. Now, substituting Eq. 3.8 and performing apartial integration, Eq. 3.7 transforms to

ρReq u(t) =

ϕ′(Req)u(t) −4G0

Req

[u(t) − u(0)e−t/λ0 −

1λ0

∫ t

0u(t ′)e(t ′−t)/λ0dt ′

].

(3.10)

Solving Eq. 3.10 is not straightforward. The exponential under the time integral suggests thetechnique of Laplace transformation to be applied. Performing a Laplace transformation

∫ ∞

0(...)e−stdt, (3.11)

from the time variable t to the new variable s (dimension 1/time) results in

ρReq

∫ ∞

0u(t)e−stdt =

(ϕ′(Req) −

4G0

Req

)∫ ∞

0u(t)e−stdt

+4G0

Requ(0)

λ0

1 + sλ0+

4G0

Req

1λ0

∫ ∞

0e−st

(∫ t

0u(t ′)e(t ′−t)/λ0dt ′

)dt .

(3.12)

The double integral of Eq. 3.12 is rewritten into

I =∫ ∞

0e−ste−t/λ0

(∫ t

0u(t ′)et ′/λ0dt ′

)dt, (3.13)

where we can change the order of integration as follows

I =∫ ∞

0u(t ′)et ′/λ0

(∫ ∞

t ′e−ste−t/λ0dt

)dt ′, (3.14)

24 Chapter 3

since u(t) is a smooth function. This equation can be rewritten into

I =λ0

sλ0 + 1

∫ ∞

0u(t ′)e−st ′dt ′. (3.15)

Now substituting Eq. 3.15 into Eq. 3.12, u(t ′) and u(t) are transformed to the Laplace domainas follows

u(s) =∫ ∞

0e−st ′u(t ′)dt ′ =

∫ ∞

0e−stu(t)dt . (3.16)

The full equation for u(s) reads

ρReq

(s2u(s) − su(0) − u(0)

)=

ϕ′(Req )u(s) −4G0

Req

(u(s) − u(s)

1sλ0 + 1

− u(0)λ0

sλ0 + 1

),

(3.17)

where the left hand side is obtained by (successive) partial integration. In the case of a multi-mode Maxwell model, the term in the right hand side starting with 4G0/Req is incorporatedin a summation over i from 1 to N , where N is the total number of considered modes. Aderivation is presented in Appendix B.Solving Eq. 3.17 for u(s), one finds

u(s) =4G0u(0) + R2

eqρu(0)λ0

+ R2eqρ

[u(0)λ0

+ u(0)]

s + R2eqρs2u(0)

−Reqϕ′(Req )

λ0+[4G0 − ϕ′(Req)Req

]s + R2

eqρ

λ0s2 + R2

eqρs3. (3.18)

As u(s) is a ratio of polynomials, its inverse transform will be a sum of exponentials describ-ing a combination of exponential decay(s) and oscillation(s) of which the time scales are setby the eigenfrequencies. At this point, we note that the numerator in Eq. 3.18 is only of limi-ted value. Since we are performing a small-amplitude expansion we are really studying thefinal or asymptotical stage of approach towards equilibrium. As a result, the initial conditionsu(0) and u(0) are not that relevant as they refer to the initial regime. In this regime non-lineareffects cannot be neglected. The denominator, however, contains the more important infor-mation as it describes the relevant time scales, via the so-called eigenfrequencies, present inthe system.Knowing this, we will now analyse the eigenfrequencies for Maxwell material behavior inmore detail. To simplify the denominator of Eq. 3.18 we introduce a dimensionless s∗ = sλ0to obtain

−ϕ′(Req)

ρReqλ2

0 +[

4G0λ20

ρR2eq

−ϕ′(Req)λ2

0ρReq

]s∗ + (s∗)2 + (s∗)3 = 0, (3.19)

where we can condense all dependencies to two dimensionless variables, X and Y

X = −ϕ′(Req)λ2

0Reqρ

,

Y =4λ2

0G0

R2eqρ

=4λ0η0

R2eqρ

,

(3.20)


with X, Y > 0. Subsequently, Eq. 3.19 can be rewritten into

X + (X + Y )s∗ + (s∗)2 + (s∗)3 = 0. (3.21)

This is a particular interesting result, since we managed to simplify our initial problem, anintegro-differential equation, to a cubic equation. In case of a multimode Maxwell modelthe polynomial is of the order (N + 2), where N is the total number of considered modes(Appendix B).In the following section (3.2.1) we simplify the cubic equation towards more tractable qua-dratic equations by considering the limits of X and Y in a quadratic analysis.

3.2.1 Quadratic analysisThe cubic equation (Eq. 3.21) has either three real roots or one real and two complex conju-gated roots. The real root is the time scale for viscous growth and the complex conjugatedroot is the time scale for elastic oscillation. In this section we will consider four limitingsituations as given in Figure 3.1.

PSfrag replacements

X → ∞Y → 0

X → 0Y → ∞ X → ∞

Y → ∞

X → 0Y → 0

1st 2nd

3rd 4th

Figure 3.1: Schematic representation of the four quadrants considered.

Using a first-order term-by-term expansion results in simplified solutions for each quadrant.

First quadrant

Here Y → ∞ while X → 0, therefore we first expand towards the large variable Y , ass0 = CY n and obtain the exponent n and pre-factor C from matching the highest powers. Forthe considered case we find

X + CY n+1 + C XY n + C2Y 2n + C3Y 3n = 0. (3.22)

The important orders in Y are given by n = 1 and n = 12 respectively. For n = 1, Eq. 3.22

simplifies to C3Y 3 = 0, C = 0 and thus s∗0 = 0. The first non-trivial term is then obtained

via the following expansion

s∗ = s∗0 +

s∗X

Y. (3.23)

26 Chapter 3

Substituting this equation in combination with s∗0 = 0 into Eq. 3.21 leads to:

X + (X + Y )s∗

XY

+(

s∗X

Y

)2

+(

s∗X

Y

)3

= 0, (3.24)

which, considering Y � X , simplifies to

s∗X = −X. (3.25)

Now consider n = 12 , Eq. 3.22 simplifies to C = ±i , thus s∗

0 = ±i√

Y . To find the first non-trivial term, s∗

0 in combination with Eq. 3.23 is substituted into Eq. 3.21. After simplificationthis leads to

s∗X = −

12

Y ± i12

√Y (X − 1). (3.26)

Thus the solutions for Eq. 3.21 in quadrant 1 simplify to

s∗ = −XY

∧ s∗ = −12

± i(√

Y +X − 1

2√

Y

). (3.27)

Second quadrant

In the second quadrant X and Y → ∞ we can rewrite Eq. 3.21 as follows

R cos θ + R(cos θ + sin θ)s∗ + (s∗)2 + (s∗)3 = 0, (3.28)

with Y = R sin θ and X = R cos θ . We first seek in the large variable R as s∗0 = C Rn and

obtain the exponent n and pre-factor C from matching the highest powers. For the case underconsideration we find n = 1 and n = 1

2 , which lead to

s∗0 = 0 ∧ s∗

0 = ±i√

R(cos θ + sin θ), (3.29)

respectively. To find the non-trivial term for both solutions we use the following expansion

s∗ = s∗0 + s∗

θ Y. (3.30)

Substitute Eq. 3.29 and the first-order expansion as given in Eq. 3.30 into Eq. 3.28 results in

s∗ =−X

X + Y∧

s∗ = −12

YX + Y + 1

± i(√

X + Y −12

Y√

X + Y (X + Y + 1)

),

(3.31)

respectively.


Third quadrant

In this quadrant we can again rewrite Eq. 3.21 as Eq. 3.28, since X and Y → 0.We first seek in the small variable R as s∗

0 = C Rn and obtain the exponent n and pre-factorC from matching the lowest powers. For the case under consideration we find n = 0 andn = 1

2 , which lead to

s∗0 = −1 ∧ s∗

0 = ±i√

R cos θ = ±i√

X, (3.32)

respectively. Substitute Eq. 3.32 and the first-order expansion as given in Eq. 3.30 intoEq. 3.28 results in

s∗ = −1 +1

1 + X + Y∧

s∗ =−2XY

4X2 − 4XY + Y 2 + 4X

±i√

X

(1 +

2XY − Y 2

4X2 − 4XY + Y 2 + 4X

),

(3.33)

respectively.

Fourth quadrant

The fourth quadrant where X → ∞ while Y → 0, we find s∗0 by substituting s∗ = s∗

0 intoEq. 3.21 and simplifying for X � Y :

(s∗0 )3 + (s∗

0 )2 + Xs∗0 + X = 0, (3.34)

which can be solved directly into

s∗0 = −1 ∧ s∗

0 = ±i√

X . (3.35)

The first non-trivial term is then obtained via the following expansion

s∗ = s∗0 + s∗

X Y. (3.36)

Substituting the results for s∗0 and Eq. 3.36 into Eq. 3.21 gives the both results for s∗

s∗ = −1 +Y

X + 1∧ s∗ = −

12

YX + 1

± i√

X(

1 +12

YX + 1

). (3.37)

The results for the four quadrants are summarized in Table 3.1.With these solutions we can find the time scales for expansion. These resulting time scales areverified using a numerical analysis (Chapter 4) and vice versa. Another verification is done bycomparing the asymptotical analysis for the Maxwell model with the asymptotical analysisusing the two limiting situations of the Maxwell model, namely the Newtonian (λ0 → 0) andHookean (λ0 → ∞) model. For both limiting situations, the numerically and asymptoticallyfound time scales are compared.

28 Chapter 3

Table 3.1: The summarized roots of Eq. 3.21 calculated in the four limits as presented in Figure 3.1.

1 X → 0, Y → ∞

s∗ = −XY

s∗ = −12

± i(√

Y +X − 1

2√

Y

)

2 X, Y → ∞

s∗ = −X

X + Y

s∗ = −12

YX + Y + 1

± i(

−12

Y√

X + Y (X + Y + 1)+

√X + Y

)

3 X, Y → 0

s∗ = −1 +Y

1 + X + Y

s∗ = −2XY

4X2 − 4XY + Y 2 + 4X± i

√X

(1 +

2XY − Y 2

4X2 − 4XY + Y 2 + 4X

)

4 X → ∞, Y → 0

s∗ = −1 +Y

X + 1

s∗ = −12

YX + 1

± i√

X(

1 +Y

2(X + 1)

)

3.2.2 NewtonianAs a starting point we used Eq. 2.25, which is the momentum equation incorporating New-tonian material behavior. Directly from this equation, using R(t) = Req + u(t), Eq. 3.8 andneglecting all u(t) contributions higher than linear order, we find

−ϕ′(Req)

ρRequ(t) +

4η0

R2eqρ

u(t) + u(t) = 0, (3.38)

which contains the viscous time scales for expansion. Considering now the denominator ofEq. 3.18 for the limit λ0 → 0 results in

−ϕ′(Req)

Reqρ+

4η0

ρR2eq

s + s2 = 0, (3.39)

which matches Eq. 3.38. Applying the definitions for X and Y to this equation leads to

X + Y s∗ + (s∗)2 = 0. (3.40)

Taking into account the expressions for X and Y as given in Eq. 3.20 with λ0 = η0/G0. Tochange to the Newtonian model, we can say that G0 → ∞ and therefore λ0 → 0 and thus

X → 0,

Y → 0.(3.41)


This implies that the solutions for the time scales as defined for the Newtonian model asgiven in Eq. 3.38 should match the results for the time scales in the quadrant 3. To check thisrelation for Newtonian material behavior, we write the roots of Eq. 3.40 as

s∗ = −12

Y ±12

√Y 2 − 4X . (3.42)

We can match this solution with the second solution found in quadrant 3 (X, Y → 0)

s∗ ≈ −12

Y ± i√

X

(1 +

12

Y −14

Y 2

X

), (3.43)

where i√

X(

1 + 12Y − 1

4Y 2

X

)equals 1

2

√Y 2 − 4X for X, Y → 0.

3.2.3 HookeanAs a starting point we used Eq. 2.26, the momentum equation incorporating the Hookeanmodel. Directly from this equation, using R(t) = Req + u(t), Eq. 3.8, Eq. 3.3 and neglectingall u(t) contributions higher than linear order, we find

−ϕ′

H (Req)

ρRequ(t) + u(t) = 0, (3.44)

where ϕ′H (Req) = ϕ′(Req) − 4G0

Req. This equation contains the elastic time scales for expan-

sion. Considering now the denominator of Eq. 3.18 for the limit λ0 → ∞ results in

−ϕ′(Req)

Reqρ+

4G0

ρR2eq

+ s2 = 0, (3.45)

which matches Eq. 3.44. Applying the definitions for X and Y to this equation leads to

X + Y + (s∗)2 = 0. (3.46)

Now, remember the expressions for X and Y as given in Eq. 3.20 with λ0 = η0/G0. Tochange to the Hookean model, we can say that η0 → ∞ and therefore λ0 → ∞ and thus

X → ∞,

Y → ∞.(3.47)

This implies that the solutions for the time scales as defined for the Hookean model as givenin Eq. 3.44 should match the results for the time scales in the quadrant 2. To check thisrelation, we write the roots of Eq. 3.46 as

s∗ = ±i√

X + Y . (3.48)

We can match this solution with the imaginary part of the second solution found in quadrant2 (X, Y → ∞):

s∗ ≈ ±i√

X + Y . (3.49)

30 Chapter 3

3.3 Values for X and Y

With the values given in Table 2.2, X and Y can be calculated. The values are given inTable 3.2 and Table 3.3 for the whole temperature range and in Table 3.4 for T = 420Kusing different initial fill fractions (α). From these tables, it can be seen that the whole rangeof X

[10−3 → 1011] and Y

[10−2 → 1011] is sampled and therefore the quadratic analysis

could not be limited to a single, simple limit e.g. X, Y → 0.

Table 3.2: Calculated variables according to the values in Table 2.2 for the different temperatures con-sidering the Newtonian/Maxwell model and the suggested quadrant Q.

T Req/R0 ϕ′(Req) X Y Q[K ] [−] [N/m3] [−] [−] [−]400 12.27 −2.44 · 104 2.95 · 1010 1.11 · 1011 2405 12.32 −2.43 · 104 6.94 · 106 2.59 · 107 2410 12.37 −2.42 · 104 8.09 · 103 2.99 · 104 2415 12.42 −2.42 · 104 3.08 · 101 1.13 · 102 2420 12.47 −2.41 · 104 2.87 · 10−1 1.04 · 100 2425 12.52 −2.40 · 104 5.37 · 10−3 1.93 · 10−2 3

Table 3.3: Calculated variables according to the values in Table 2.2 for the different temperatures con-sidering the Hookean model and the suggested quadrant Q.

T Req/R0 (H ) ϕ′(Req) X Y Q[K ] [−] [N/m3] [−] [−] [−]400 6.39 −3.33 · 105 7.74 · 1011 4.11 · 1011 2405 6.43 −3.29 · 105 1.80 · 108 9.51 · 107 2410 6.47 −3.24 · 105 2.07 · 105 1.09 · 105 2415 6.52 −3.19 · 105 7.74 · 102 4.09 · 102 2420 6.56 −3.15 · 105 7.15 · 100 3.77 · 100 2425 6.60 −3.11 · 105 1.32 · 10−1 6.96 · 10−2 3

3.4 Results: Asymptotical versus numerical time scales

3.4.1 Newtonian

Using the asymptotical analysis for Newtonian material behavior, described in Eq. 3.39, wecan estimate the time scales for expansion (s).From the figures presented in Chapter 4, in which the numerical determined growth of thesingle bubble vs. time for Newtonian material behavior is shown, we can determine the time


Table 3.4: Calculated values according to Table 2.2 for different fill fractions (α) at T = 420K consi-dering the Newtonian/Maxwell model and the suggested quadrant Q.

α Req/R0 ϕ′(Req) X Y Q[−] [−] [N/m3] [−] [−] [−]1.0 12.47 −2.41 · 104 2.87 · 10−1 1.04 · 100 20.8 11.58 −2.59 · 104 3.33 · 10−1 1.21 · 100 20.6 10.52 −2.85 · 104 4.04 · 10−1 1.46 · 100 20.4 9.19 −3.26 · 104 5.29 · 10−1 1.92 · 100 20.2 7.29 −4.11 · 104 8.39 · 10−1 3.04 · 100 2

scales for expansion by plotting E as function of t , where

E(t) =Req − R(t)

Req= −

u(t)Req

, (3.50)

which is related to the time scale 1/τ by

E(t) ∝ exp(−tτ). (3.51)

The time scales for both the asymptotical and the numerical analysis are presented in Fi-gure 3.2 and in Table 3.5.

400 405 410 415 420 42510

−6

10−5

10−4

10−3

10−2

10−1

100

101

PSfrag replacements

T [K ]

s[1/

s],1

/τ[1

/s]

Figure 3.2: The time scales for expansion, ’solid line’ s [1/s] obtained from the asymptotical analysisand the time scales for expansion, ’◦’ 1/τ [1/s] obtained from the actual single bubbleexpansion for Newtonian behavior (Chapter 4) as function of temperature.

Figure 3.2 shows that s and 1/τ match almost perfectly for the whole temperature range. Wehave to emphasize that these time scales are independent of the initial conditions, since weare considering an asymptotical analysis, implying that we start near the equilibrium radius.From this we can conclude that the initial conditions are not relevant for the time scales ofexpansion.

32 Chapter 3

Table 3.5: The time scales for expansion (s) obtained from the asymptotical analysis and the numericalresult (1/τ ) obtained in Chapter 4, including the relative error, based on the asymptoticvalue.

T [K ] s[1/s] 1/τ [1/s] rel. error400 2.18 · 10−6 3.20 · 10−6 0.47405 1.43 · 10−4 1.52 · 10−4 0.06410 4.21 · 10−3 4.36 · 10−3 0.04415 6.88 · 10−2 7.02 · 10−2 0.02420 3.30 · 10−1 3.35 · 10−1 0.02425 1.37 · 100 1.42 · 100 0.04

3.4.2 Hookean

Using the asymptotical analysis for Hookean material behavior, Eq. 3.45, we can determinethe frequency of oscillation (s).From the figures presented in Chapter 4, we can determine the numerically determined fre-quency for the (non-harmonic) oscillation (ω), which is defined by

ω =2π

T, (3.52)

where T is the period of the oscillation.Both frequencies for the whole temperature range are presented in Figure 3.3 and Table 3.6.

400 405 410 415 420 4255

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

PSfrag replacements

T [K ]

s[1/

s],ω

[1/s]

Figure 3.3: The frequencies of oscillation, ’solid line’ s[1/s] obtained from the asymptotical analysisand ’◦’ ω[1/s] obtained from numerical analysis (Chapter 4), as function of temperature,T [K ].

From Figure 3.3 it is clear that the numerical and analytical solution show the same trendin the complete temperature range. Which is surprising, since we did not expect that usingEq. 3.8 to find an expression for ϕ(Req + u(t)) was allowed for non-harmonic, undampedoscillations.


Table 3.6: The frequencies for oscillation (1/s) obtained from the asymptotical analysis and the fre-quencies for oscillation (ω) obtained from numerical analysis as presented in Chapter 4 forHookean material behavior as function of temperature, including the relative error, based onthe asymptotic value.

T [K ] s[1/s] ω[1/s] rel. error400 8.94 6.55 0.27405 8.85 6.48 0.27410 8.75 6.40 0.27415 8.64 6.31 0.27420 8.56 6.25 0.27425 8.48 6.18 0.27

To find this discrepancy and to verify the origin of it, we analyzed our momentum equationincorporating the Hookean model in more detail. We used the data corresponding to T =410K . The left plot of Figure 3.4 shows the numerically found radius (’◦’) and a harmonicoscillation (’solid line’) given by

R(t) = Req +(R0 − Req

)cos (ωt) , (3.53)

using the numerically found ω = 6.40 and Req/R0 = 6.47. The right hand side shows thatthe numerical value of ϕ(Req +u(t))−ϕ(Req) overlaps with the asymptotical assumed valueϕ′(Req) only for R/R0 → Req/R0 = 6.47.

0 0.5 1 1.5 2 2.50

2

4

6

8

10

12

14

PSfrag replacements

t[s]

R/

R0

0 0.5 1 1.5 2 2.5

x 106

0

2

4

6

8

10

12

14

PSfrag replacements

ϕ′(Req)u(t), ϕ(Req + u(t)) − ϕ(Req)

R/

R0

Figure 3.4: Left: The numerically found radius R/R0(t) (’◦’) and a harmonic oscillation R/R0(t)(’solid line’) equal to Eq. 3.53 with Req/R0 = 6.47 and ω = 6.40 (all considered datacorrespond to T = 410K ). Right: The radius R/R0 as function of the numerical usedvalue for ϕ(Req + u) − ϕ(Req) (’◦’) and the asymptotically assumed value for calculationϕ′(Req )u(t) (’solid line’).

Would we now consider a more simple expression for ϕ equal to

ϕ(R) = c[R(t) − Req ], (3.54)

34 Chapter 3

with c = ϕ′(Req) = −3.24 · 105N/m3 (corresponding to the value found in Table 3.3 forT = 410K ) results in Figure 3.5. The adoption of ϕ(R) = c[R(t) − Req ] explicitly meansthat at R(t) = Req the potential force for expansion is zero. For this reason, the amplitudeof the numerically found oscillation is at its maximum at R(t) = Req . The right plot ofFigure 3.5 shows that the assumption made in Eq. 3.8 is valid for the adopted ϕ(R).

0 0.5 1 1.5 20

2

4

6

PSfrag replacements

t[s]

R/

R0

0 0.5 1 1.5 2

x 106

0

2

4

6

PSfrag replacements

ϕ′(Req)u(t), ϕ(Req + u(t))R/

R0

Figure 3.5: Left: The numerical found radius R/R0(t) for ϕ equal to c[R(t) − Req ] with c =−3.24 · 105 N/m3 and Req = 6.47 (considered data correspond to T = 410K ). Right:The numerical used value for ϕ(Req + u) (’◦’) and the asymptotically assumed value forcalculation ϕ′(Req)u(t) (’solid line’).

From the left plot of Figure 3.5, we can determine the period of oscillation, which equalsT = 0.72. With this value we can calculate the frequency of oscillation, equal to ω = 8.75.This value corresponds directly to the asymptotically found value as presented in Table 3.6.Now, as a final check we consider the momentum equation as given by Eq. 2.22 and com-pletely neglect the integral gives

32

R2 + RR =κg

ρR3 −pext

ρ−

2γ

ρR. (3.55)

Considering the data as given for T = 410K , where Req/R0 can be calculated from ϕ(Req) =0 equals 12.37, and calculating R/R0(t) for different R0 (creates a different u0 since u0 =R0 − Req) as is used in the left plot of Figure 3.6. The right plot compares the asymptoticallyfound timescale s represented by a ’solid line’, which is of course no function of u0, with thenumerically found timescale ω, which is expected to be a function of u0.From Figure 3.6 it is shown that by increasing R0 towards Req (and thus decreasing thevalue of u0) the error of the calculated time scales for oscillation decrease as a result of theoscillations to tend from non-harmonic to harmonic.

3.4.3 Maxwell model

Solving the momentum equation by considering the material to behave as a linear viscoelasticmaterial (Maxwell model), we find a combination of oscillation and exponential growth ofthe single bubble.


0 5 100

5

10

15

20

25

30

PSfrag replacements

t[s]

R/

R0

0 2 4 6 8 10 121.1

1.2

1.3

1.4

1.5

PSfrag replacements

u0[−]

ω[1

/s]

,s[1

/s]

Figure 3.6: Left: The single bubble radius R/R0(t), considering no material behavior, in the directionof the arrow, the initial diameter R0, which sets the amplitude of oscillation takes the val-ues 1, 5, 10, 12, approaching the equilibrium radius for the system equal Req = 12.37.Right: The asymptotical determined timescale, s (’solid line’) and the numerical deter-mined timescale, ω (’◦’) as function of u0.

For coupling the numerical outcome with the asymptotical outcome (as found by solvingEq. 3.21) we have to consider the numerical outcome for larger time scales, for t going toinfinity. This suggests the use of (fast) Fourier transforms (FFT). We could write the radiusR(t) as

R(t) = Req + A0 exp(−t/τ0) +∞∑

j=1

(A jCτ j ,� j (t) + B j Sτ j ,� j (t)

), (3.56)

where A0, A j and B j are the fit parameters,

Cτ j ,� j (t) = e−t/τ j cos(� j t), (3.57)

and

Sτ j ,� j (t) = e−t/τ j sin(� j t). (3.58)

Performing a continuous Fourier transformation on R(t) is given by

R(ω) =∫ tb

taR(t)e−iωt dt, (3.59)

which leads to

R(ω) = Reqδ(ω) +A0

τ0+

∞∑

j=1

(A j Cτ j ,� j (ω) + B j Sτ j ,� j (ω)

), (3.60)

where

Cτ j ,� j (ω) =∫ tb

tae−t/τ j cos(� j t)e−iωt dt

=12

(1

1/τ j + i(� j + ω)+

11/τ j − i(� j − ω)

),

(3.61)

36 Chapter 3

and

Sτ j ,� j (ω) =∫ tb

tae−t/τ j sin(� j t)e−iωt dt

=12i

(1

1/τ j − i(� j − ω)−

11/τ j + i(� j + ω)

).

(3.62)

We need to consider the imaginary (=) and the real (<) parts of the Fourier transformedsolutions.

=[Cτ j ,� j (ω)

]=

12

(−(� j + ω)

(1/τ j )2 + (� j + ω)2 +� j − ω

(1/τ j )2 + (� j − ω)2

), (3.63)

<[Cτ j ,� j (ω)

]=

12

(1/τ j

(1/τ j )2 + (� j + ω)2 +1/τ j

(1/τ j )2 + (� j − ω)2

), (3.64)

=[Sτ j ,� j (ω)

]=

12

( −1/τ j

(1/τ j )2 + (� j − ω)2 +1/τ j

(1/τ j )2 + (� j + ω)2

), (3.65)

<[Sτ j ,� j (ω)

]=

12

(� j − ω

(1/τ j)2 + (� j − ω)2 +� j + ω

(1/τ j )2 + (� j + ω)2

), (3.66)

and thus

=[R(ω)

]= A0

−ω

(1/τ0)2 + ω2 +

∞∑

j=1

(A j=

[Cτ j ,� j (ω)

]+ B j=

[Sτ j ,� j (ω)

]).

(3.67)

<[R(ω)

]= δ(ω)Req + A0

1/τ0

(1/τ0)2 + ω2 +

∞∑

j=1

(A j<

[Cτ j ,� j (ω)

]+ B j<

[Sτ j ,� j (ω)

])).

(3.68)

The asymptotical analysis gives as result three solutions, one real and two complex conju-gated. This implies the use of j = 1 in Eq. 3.56. Fitting the imaginary part (Eq. 3.67) onthe FFT of the numerical outcome (Chapter 4) results in the numerical time scales as givenin Figure 3.8 and Table 3.7 as function of the fill fraction α.We can conclude from the values in Figure 3.8 and Table 3.7, that the asymptotical analysisresults in approximately the same time scales as for the numerical analysis. Where we have toemphasize that the error, which is in the order of a few percent, is a result of the data analysis,and is explained in more detail in Section 3.5.


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.57

0.58

0.59

0.6

0.61

0.62

0.63

0.64

0.65

0.66

0.67

PSfrag replacements

α[−]

1/τ 0

[1/s]

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.96

0.97

0.98

0.99

1

1.01

1.02

1.03

PSfrag replacements

α[−]

1/τ 1

[1/s]

Figure 3.7: The asymptotical (’solid line’) and numerical (’◦’) (see Chapter 4) determined viscous timescales (1/τ ) for expansion using Maxwell material behavior at T = 420K as function ofdifferent fill fractions α. Left: 1/τ0 and Right: 1/τ1.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.5

3

3.5

4

4.5

5

PSfrag replacements

α[−]

�1[

1/s]

Figure 3.8: The asymptotical (’solid line’) and numerical (’◦’) (see Chapter 4) determined elastic timescales (�1) for expansion using Maxwell material behavior at T = 420K as function ofdifferent fill fractions α.

Table 3.7: The frequencies for oscillation (�) and time scales for expansion (1/τ ) obtained from theasymptotical and numerical analysis (Chapter 4) using Maxwell material behavior at T =420K using different fill fractions α (all variables have the dimension [1/s] except α).

α[−] 1/τasymp0 1/τ

asymp1 �

asymp1 1/τ num

0 1/τ num1 �num

11.0 0.653 0.970 2.592 0.661 0.975 2.5810.8 0.637 0.978 2.855 0.628 0.987 2.8460.6 0.621 0.986 3.217 0.608 0.984 3.2480.4 0.604 0.994 3.775 0.596 1.002 3.7820.2 0.587 1.003 4.890 0.583 1.023 4.911

38 Chapter 3

3.5 Error sourceThe time scales found using our numerical and asymptotical analysis are of the same order.The errors between both values are small enough to find the right trend or to estimate the timescales for expansion of our single bubble. Where do the errors originate from:

• To find numerically accurate plots of the radius as function of time, we need to con-sider small 1t . To compare the numerically found time scales with our asymptoticallyfound time scales, our numerical analysis should continue for a very long period oftime, suggesting to incorporate a large 1t . To satisfy both demands, we have to find acompromise, decreasing the accuracy of our numerical analysis.

• For the specific case of the Hookean model we introduce errors by using the linearcontribution for ϕ

(Req + u(t)

)as proposed in Eq. 3.8. Nevertheless, the data as given

in Table 3.6 show a rather good overlap between the numerically and asymptoticallydetermined time scales.

• The prepared equation used for fitting the imaginary part of the Fourier transform ofthe numerical found radius as function of time contains a summation from j = 1 up to∞ (Eq. 3.67). We used j = 1 as a maximum. Fitting this equation introduces errors asa result of the disturbances present in our signal for j > 1.

• The data for Maxwell material behavior is achieved by fitting the Fourier transform ofthe numerical found R(t), a procedure whose errors we cannot easily quantify.

3.6 ConclusionsAn asymptotical method to find the time scales for the expansion of a single bubble in generalwas introduced. This method was used by incorporating data concerning our model system,polystyrene as shell material and water as the given core.With the asymptotical method we are able to predict the time scales for expansion concerningthe Maxwell model and to validate our numerical method and vice versa. Also for the twolimiting situations of the Maxwell model, the Newtonian and Hookean model, this is possible.Whereas, for the Hookean model, we have to emphasize that the calculated asymptoticaltime scales for oscillation have to be considered carefully, since the oscillations are non-harmonic and undamped. In real-life, expansion can show oscillations, but these will alwaysdamp, making our analysis valid as shown by decreasing the amplitude of oscillation. Thisdecrease results in an increase of the numerical determined time scale up to the asymptoticaldetermined time scale.We can conclude from the values concerning the Maxwell model, that the asymptotical ana-lysis results in approximately the same time scales as for the numerical analysis. Where wehave to emphasize that the error, which is in the order of a few percent, is a result of the dataanalysis, using the proposed fit procedure. This validates our asymptotical analysis, whichis shown to be a very powerful tool for not only predicting numerical outcome, but also forvalidating the numerical results. Another possibility would be to predetermine the time step1t for numerical calculations, decreasing the necessary calculation time.Our asymptotical analysis showed that the time scales for bubble growth are independentof the initial conditions, since this analysis is asymptotical, implying that we start near the


equilibrium radius. From this we can conclude that the initial conditions are not relevant forthe time scales of expansion.If we would go one step further, we would expect that our asymptotical analysis, with afew modifications, is applicable to a multimode Maxwell model. Unfortunately, the order ofthe polynomial to be solved increases with the number of modes N included, which makesour analysis more complex. For the single mode Maxwell model, we had to solve a third-order polynomial (cubic equation). For a N mode Maxwell model we expect to end up with a(N +2)-order polynomial. Where we have to state that in a multimode approximation usuallyone mode (N = 1) is the ’leading’ mode for which we could use our method to determinethe time scales for expansion.

40 Chapter 3

Chapter 4

Single bubble growth: Numericalanalysis

4.1 Introduction

In Chapter 2, the mathematical framework was expressed in a momentum equation governingthe single bubble growth, including several constitutive equations for describing the materialbehavior of a single bubble shell. In Chapter 3 we used this framework to determine the timescales for expansion in an asymptotical way for the Newtonian, Hookean and Maxwell model.In this chapter, also an asymptotical method for incorporating the multimode Maxwell modelwas proposed. To verify the asymptotically determined time scales, we used the numericalanalysis as presented in the current chapter and vice versa. Subsequently, we expanded ournumerical analysis with the multimode Maxwell model, the extended Pom-Pom model anddiffusion of the blowing agent to the exterior of the system (loss of blowing agent) can beincluded.A lot of literature exists on the subject of bubble growth and bubble oscillation in differentmedia. Street et al. (1971) established the initial framework for analyzing bubble growth.A finite envelope of fluid was placed around the bubble, later referred to as the single bub-ble model. They analyzed the coupled equation of motion, equation of continuity and theequation of energy for this system. Since this early work a lot of extensions have been pub-lished. These works included improved cell descriptions and more realistic conditions at theouter cell boundary (Amon and Denson, 1984), inclusion of viscoelastic effects (Arefmanesh,1991), analysis of bubble growth during process flow (Amon and Denson, 1986; Arefmaneshet al., 1990), comparison of numerical foam expansion results with experiments (Rameshet al., 1991), simultaneous nucleation and bubble growth (Joshi et al., 1998) and complexrheological behavior (Agarwal, 2002).The objective of this chapter is to develop a general framework in which complex rheologi-cal behavior can be taken into account, e.g. including multimode and non-linear constitutiveequations. Our numerical analysis is also used to verify the asymptotical analysis, as descri-bed in Chapter 3, and vice versa. In Chapter 6, the computational results are compared tothe experimental expansion characteristics of water expandable polystyrene, as described inChapter 5.

42 Chapter 4

First of all we rewrite the equations as presented in Chapter 2, the conservation, the consti-tutive and the diffusion equations, into their dimensionless form, which is useful in the restof our computational analysis. Subsequently, we rewrite the (in Chapter 2) proposed inputparameters into dimensionless values. After this, we present the computational techniquesused for solving the proposed dimensionless equations per constitutive model, since everymodel demands a suitable computational technique. Also the momentum equation in combi-nation with the diffusion demands a suitable computational technique and is given hereafter.In the results section, we show the single bubble radius as function of time for the differentconstitutive equations using the in Chapter 2 proposed parameters for the used model system(polystyrene as shell material and water, the blowing agent, as core material, later referredto as WEPS) as a starting point. We end up by showing the influence of the diffusion on thesingle bubble growth.

4.2 Dimensionless equations

4.2.1 Conservation and constitutive equationsThe expansion of a single bubble in a large body of incompressible fluid is given by Eq. 2.22,using the integral form of the Maxwell material behavior constitutive equation. To estimatethe importance of the various characteristic terms in this equation which influence the growthof the bubble, this equation is transformed into a dimensionless form

3292 + 99 =

κ∗g

93 − 1 −2

NW e9−

12NDe NRe

∫ t∗

0e

−t∗+t1NDe

91921 ln

[919

]

931 − 93

dt1, (4.1)

where the Deborah, the Reynolds and the Weber number are defined as

NDe =λ0

tc; NRe =

ρR20

η0tc; NW e =

ρR30

t2c γ

,

respectively, and 9 = R/R0, 9 = d9/dt , 9 = d29/dt2, t∗ = t/tc, 91 = 9 at a previoustime t1, κ∗

g = κg/(R30 pext) and tc = R0(ρ/pext)

1/2 is a characteristic expansion time. Theinitial conditions are given by 9(0) = 1 and 9(0) = 0, where the latter one is the result ofthe system being at rest at t = 0.The Deborah number is the ratio of characteristic relaxation time of the polymer and charac-teristic flow time, the Reynolds number is the ratio of inertial force to viscous force and theWeber number is the ratio of inertial force to surface tension force.In Chapter 2, we mentioned the radial velocity vr at radial position r in the fluid, whichis irrespective of the detailed fluid rheology and is a result of the equation of continuityincorporating spherical symmetry (Eq. 2.3) as applicable for the single bubble model. Thisequation is transformed into a dimensionless equation

v∗r =

929

(r∗)2 , (4.2)

where the dot denotes the time derivative with 9 = d9/dt∗.

Single bubble growth: Numerical analysis 43

The differential form of the Maxwell material behavior description (Eq. 2.19) can also berewritten using the above mentioned dimensionless quantities

NDedτ

∗

dt∗+ τ

∗ =2D∗

NRe, (4.3)

where τ∗ = τ/pext and D∗ = Dtc. This equation can be solved in combination with the

dimensionless form of Eq. 2.10 which is given by

3292 + 99 =

κ∗g

93 − 1 −2

NW e9+∫ ∞

9

(2τ ∗

rrr∗ −

τ ∗θθ + τ ∗

φφ

r∗

)dr∗. (4.4)

The momentum equation describing the single bubble growth using Newtonian material be-havior (Eq. 2.25) can be rewritten using the earlier introduced dimensionless quantities to

3292 + 99 =

κ∗g

93 − 1 −2

NW e9−

49

NRe9. (4.5)

The momentum equation describing the single bubble growth using Hookean material beha-vior (Eq. 2.26) can be rewritten using the same dimensionless quantities to

3292 + 99 =

κ∗g

93 − 1 −2

NW e9−

12NRe NDe

∫ t∗

0

91921 ln

[919

]

931 − 93

dt1. (4.6)

The multimode approximation is often necessary for a realistic description of the viscoelasticcontributions. The correlation between the fluid-phase stresses for a multimode Maxwellmodel (Eq. 2.27) can be rewritten using the earlier introduced dimensionless quantities to

(NDe)idτ

∗i

dt∗+ τ

∗i =

2D∗

(NRe)i, (4.7)

where the viscoelastic contribution of the i th relaxation mode is denoted by τ∗i .

To capture the non-linear behavior in both elongation and shear, we used the extended Pom-Pom (XPP) model (Verbeeten et al., 2001). With this constitutive equation we are able toinvestigate the e.g. strain hardening effect on the bubble growth. Where we have to notethat for WEPS hardly any strain hardening effect is expected, since the applied polystyrene isgenerally linear with a low molar mass. Also for the XPP model a multimode approximationof the relaxation spectrum is necessary. Below the constitutive behavior for a single mode ofthe viscoelastic contribution is rewritten into dimensionless quantities

∇τ

∗=

2D∗

NRe NDe− λ

∗(τ∗)−1 · τ∗,

λ∗(τ∗)−1 =

1λ∗

0b

[αNRe NDeτ

∗ + f ∗(τ ∗)−1 I +( f (τ ∗)−1 − 1)

τ ∗NRe NDe

],

f ∗(τ ∗)−1 =2λ∗

0bλ∗

s

(1 −

13∗

)+

1

(3∗)2

[1 −

13α I ∗

τ ·τ N2Re N2

De

],

3∗ =√

1 +13

I ∗τ NRe NDe,

λ∗s = λ∗

0se−ν(3∗−1),

(4.8)

44 Chapter 4

with I is the unit tensor and where∇τ

∗= tc

∇τ /pext , λ

∗(τ ∗)−1 = tcλ(τ )−1, λ∗0b = λ0b/tc =

NDe, λ∗0s = λ0s/tc, I ∗

τ = Iτ /pext , I ∗τ ·τ = Iτ ·τ /p2

ext and ν = 2/q with q the number of armsat the end of a backbone.

4.2.2 Mass diffusionThe diffusion of the blowing agent to the exterior of the bead is of major importance (seeChapter 5) on the expansion process and is independent of the material behavior of the poly-mer. To combine the momentum equation and the diffusion of the blowing agent, we choseour polymer shell to behave as a Newtonian material. Including mass diffusion in combina-tion with the momentum equation incorporating the Newtonian model needs a slight changecompared to Eq. 4.5 and is rewritten into

3292 + 99 = p∗

g − 1 −2

NW e9−

49

NRe9, (4.9)

with

p∗g =

pg

pext=

mg(t)RgT

MB A43π R(t)3 pext

, (4.10)

where mg is the mass of the gas present in the bubble and the initial value equal to m0g =

m0g = αρB A

43π R3

0 . Now, rewriting Eq. 2.39 in the necessary dimensionless format results in

dmg

dt∗=

43π tc

692ρB A D(c0 − kh pg

)

δ. (4.11)

Combining the momentum equation incorporating the Newtonian model and the Eq. 4.11 isexpected to show after the increase in volume as function of time a decrease, which is referredto as collapse. The dependence of this collapse as function of the diffusion coefficient isinvestigated.

4.3 Dimensionless input parametersThe dimensionless variables derive directly from the in Section 2.6 proposed parameters. Forthe Maxwell model and both limit situations the dimensionless parameters are presented inTable 4.1.For the multimode Maxwell model, we use the fits as presented in Section 2.6. From thesevalues, we can calculate the mode dependent dimensionless variables, as presented in Ta-ble 4.2.To capture the non-linear behavior in both elongation and shear, we used the extended Pom-Pom (XPP) model. In this model the relaxation spectrum is required. This is representedby the multimode Maxwell model. For which also the multimode approximation of therelaxation spectrum is used as presented in Table 4.2. To investigate the effect of strainhardening on the bubble growth, we proposed the non-linear parameters as presented in Ta-ble 4.3 for N = 6. Increasing the value for q, e.g. from q1 to q2 with constant λ∗

0b,i/λ∗0s,i

(λ∗0b,i = (NDe)i ) shows an increase in strain hardening. The α is set at zero, which is a result

of the used deformation tensor D∗.


Table 4.1: The values of the proposed dimensionless quantities derived from Section 2.6 with tc thecharacteristic expansion time equal to 0.1.

T [K ] κ∗g NDe NRe NW e

400 1.85 · 103 1.22 · 106 2.91 · 10−7 3.03 · 106

405 1.87 · 103 1.87 · 104 1.91 · 10−5 3.06 · 106

410 1.89 · 103 6.43 · 102 5.62 · 10−4 3.10 · 106

415 1.92 · 103 3.98 · 101 9.16 · 10−3 3.13 · 106

420 1.94 · 103 3.86 · 100 9.53 · 10−2 3.17 · 106

425 1.96 · 103 5.30 · 10−1 7.00 · 10−1 3.21 · 106

Table 4.2: Dimensionless parameters for the multimode Maxwell model for different number of modesN at T = 410K derived from Section 2.6 with tc = 0.1.

N (NDe)i (NRe)i

1 2.66 · 102 1.37 · 10−2

1 2.06 · 10−3 1.41 · 101

2 6.68 · 103 3.63 · 10−3

1 1.04 · 10−3 2.01 · 101

2 5.12 · 10−1 2.15 · 100

3 1.70 · 102 2.43 · 10−2

4 5.34 · 103 4.90 · 10−3

1 8.54 · 10−4 2.33 · 10−5

2 1.11 · 10−1 6.12 · 10−2

3 4.73 · 100 8.98 · 10−1

4 1.01 · 102 5.76 · 101

5 1.06 · 103 1.11 · 102

6 9.66 · 103 6.90 · 102

Table 4.3: Non-linear parameters for the extended Pom-Pom model for N = 6 at T = 410K with(λ∗

0b)i = (NDe)i .

N (q1)i (q2)i λ0b,i/λ0s,i

1 1.0 1.0 62 1.0 1.0 53 1.5 4.0 44 1.5 4.0 35 2.0 10.0 26 2.0 10.0 2

46 Chapter 4

4.4 Computational techniques

In this section is explained how the dimensionless equations, Eq. 4.2 - Eq. 4.11, are imple-mented in the numerical scheme. In the previous chapter we considered several constitutiveequations and every constitutive model in combination with the proposed momentum equa-tion requires a suitable numerical strategy.We will start using the Newtonian model after which the Hookean, Maxwell, multimodeMaxwell and extended Pom-Pom model will follow respectively. In all used systems a timeand spatial discretization is necessary, except for the Newtonian and the Hookean model.In both cases only a time discretization is necessary in combination with information at theboundaries of the single bubble.

4.4.1 Newtonian

In case of Newtonian material behavior we rewrite Eq. 4.5 into two ordinary differentialequations

ddt∗

9 = 9,

ddt∗

9 = 9 =19

[κ∗

g

93 − 1 −2

NW e9−

49

NRe9−

3292]

,

(4.12)

where the two only unknowns are 9 and 9. These equations are solved in a coupled way andare discretized in time using simple first-order Euler forward results in

{9n+1 = 1t∗9n + 9n,

9n+1 = 1t∗9n + 9n,(4.13)

where 9n and 9n denote the dimensionless radius 9 and the dimensionless time derivativeof the radius 9, as denoted in Eq. 4.12, at time t = n1t respectively. For 9n = 9(t = n1t)we use the expression as denoted in Eq. 4.12. The initial conditions read

{90 = 9(t∗ = 0) = 1,

90 = 9(t∗ = 0) = 0,(4.14)

since the initial radius 90 is the radius R(t∗ = 0) scaled with R0 and the derivative of thedimensionless radius 9 equals zero at t∗ = 0 since the bubble is initially at rest.The time step 1t∗ is dependent on κ∗

g , NW e, NRe and the applied 1t∗ is described in theresults section.

4.4.2 Hookean

The momentum equation incorporating Hookean material behavior is given by Eq. 4.6. Forthis constitutive equation we need to consider the history of strain or as rewritten in this case


the history of the radius 9n1 = 9(t∗ = (n − 1)1t∗). Eq. 4.6 is rewritten as a set of three

ordinary differential equations

ddt∗

9 = 9,

ddt∗

9 = 9 =19

[κ∗

g

93 − 1 −2

NW e9−

12NRe NDe

Itot −3292]

,

ddt∗

Itot =∫ ∞

9

921 91

(r∗)4 + 931 − 93

dr∗,

(4.15)

with r∗ = r/R0 and I is introduced to eliminate the time integral in Eq. 4.6. Discretizingthese equations in time using simple first-order Euler forward leads to

9n+1 = 1t∗9n + 9n,

9n+1 = 1t∗9n + 9n,

I n+1 = 1t∗ I n + I n,

(4.16)

where I n and I n denote the dimensionless integral and the dimensionless time derivative ofthe integral at time t = n1t respectively as defined in Eq. 4.15. The integral in Eq. 4.16 canbe solved by the trapezoidal rule. The initial conditions are given by

90 = 9(t∗ = 0) = 1,

90 = 9(t∗ = 0) = 0,

I 0tot = I (t∗ = 0) = 0,

(4.17)

where 90 and 90 are the same values as for the Newtonian model and I 0tot = 0 since the

bubble is initially at rest.The time step 1t∗ is dependent on the dimensionless variables (κ∗

g , NW e, NRe , NDe), theapplied 1t∗ is described in the results section.

4.4.3 Maxwell model

In case of the Maxwell model, we again need to consider the history of strain. For theHookean model, we were able to rewrite this as a history of the radius of the bubble. However,for the Maxwell model, this is not possible and therefore we need to consider not only a timediscretization but also a spatial discretization. The momentum equation as given by Eq. 4.4

48 Chapter 4

can be rewritten into two ordinary differential equations

ddt∗

9 = 9,

ddt∗

9 = 9 =19

[κ∗

g

93 − 1 −2

NW e9+∫ S∗

9

(2τ ∗

rr

r∗ −2τ ∗

θθ

r∗

)dr∗

],

=19

[κ∗

g

93 − 1 −2

NW e9+ Itot

],

v∗r =

929

(r∗)2 .

(4.18)

The linear viscoelastic material behavior (Eq. 2.19) is given by a first-order differential equa-tion

ddt∗

τ∗ =

2NRe NDe

D∗ +τ

∗

NDe. . (4.19)

Eq. 4.18 and Eq. 4.19 must be solved in a coupled way. Discretizing Eq. 4.18 in time usinga combination of simple first-order Euler forward and the trapezoidal rule to obtain the sum-mation for Itot yields:

9n+1 = 1t∗9n + 9n,

9n+1 = 1t∗9n + 9n,

I n+1tot = I n

tot +∑

j

r∗,n( j + 1) − r∗,n( j)2

(2(τ ∗

rr )n( j)−2(τ ∗θθ)

n( j)r∗,n ( j) + 2(τ ∗

rr )n( j+1)−2(τ ∗

θθ)n( j+1)

r∗,n ( j+1)

),

(4.20)

where j represents the spatial discretization of the bubble shell. Initially, an equidistantdistribution of nodes is placed in the bubble shell. During the time integration, the positionof the spatial nodes is updated in a Lagrangian sense using the following equation

r∗,n+1 = r∗,n + v∗,nr 1t∗. . (4.21)

As a result of the Lagrangian update of the nodes, it is possible to directly discretize thematerial derivative of the stress tensor. Discretizing the constitutive equation in time usingsimple first-order Euler backward leads to

(τ ∗rr )

n =1t∗

NDe + 1t∗

[−4NRe

9n(9n)2

(r∗,n)3 +NDe(τ

∗rr )

n−1

1t∗

],

(τ ∗θθ )

n =1t∗

NDe + 1t∗

[2

NRe

9n(9n)2

(r∗,n)3 +NDe(τ

∗θθ )

n−1

1t∗

].

(4.22)

We used the same initial conditions as used for Hookean constitutive equation in combinationwith {

τ ∗rr (0) = τrr (t∗ = 0) = 0,

τ ∗θθ (0) = τθθ (t∗ = 0) = 0,

(4.23)

since the initial body of fluid is at rest.


4.4.4 Multimode Maxwell

We can use the momentum equation (Eq. 4.4) as used for the simple Maxwell model, whichcan be described into Eq. 4.18. The multimode Maxwell model (Eq. 4.7) can be rewritteninto a first-order differential equation including a summation over the number of consideredmodes N

ddt∗

τ∗i =

2(NRe)i (NDe)i

D∗ +τ

∗i

(NDe)i,

τ∗ =

N∑

i=1

τ∗i ,

(4.24)

where the viscoelastic contribution of the i th relaxation mode is denoted by τ∗i and where N

is the total number of different modes.Discretizing a single mode of the multimode Maxwell equation using simple first-order Eulerbackward yields Eq. 4.22. After each time step a summation over the stresses has to beperformed as shown in Eq. 4.24.As initial conditions we used the same as used for the single mode Maxwell model.

4.4.5 Extended Pom-Pom model

The coupling of spatial and time discretization as used for the simple Maxwell and multimodeMaxwell model can also be used for more complex constitutive equations, e.g. extendedPom-Pom model (XPP). For this constitutive model, we can use the momentum equation(Eq. 4.4) as used for the simple Maxwell model, which can be written into Eq. 4.18, whereasthe extended Pom-Pom model (Eq. 4.8) can be rewritten into

∇τ

∗i =

2D∗

(NRe)i (NDe)i− λ

∗i (τ

∗i )

−1 · τ∗i ,

τ =N∑

m=1

τ i .

(4.25)

Discretizing the XPP model for a single mode using simple first-order Euler backward yields

(τ ∗

rr)n =

2D∗,nrr 1t∗

NRe NDe+(τ ∗

rr)n−1

1 +(λ

∗,nrr)−1

1t∗ − 2L∗,nrr 1t∗

,

(τ ∗θθ

)n =2D∗,n

θθ 1t∗

NRe NDe+(τ ∗θθ

)n−1

1 +(λ

∗,nθθ

)−11t∗ − 2L∗,n

θθ 1t∗,

(4.26)

50 Chapter 4

where

D∗,nrr = L∗,n

rr =−29n (9n)2

(r∗,n)3 ,

D∗,nθθ = L∗,n

θθ =9n (9n)2

(r∗,n)3 ,

(λ∗,n

rr)−1 = 1

λ∗0b

(α(τ ∗

rr)n NRe NDe +

(f (τ )−1)n +

(f (τ )−1)n−1

NRe NDe(τ ∗rr )

n

),

(λ

∗,nθθ

)−1 = 1λ∗

0b

(α(τ ∗θθ

)n NRe NDe +(

f (τ )−1)n +(

f (τ )−1)n−1NRe NDe(τ ∗

θθ )n

),

(4.27)

with

(f (τ )−1

)n= 2λ∗

0bλ∗

s

n (1 − 1

(3∗)n

)+(

1(3∗)n

)2 (1 − 1

3α I ∗,nτ ·τ N2

Re N2De

),

(λ∗

s)n = λ0se−ν((3∗)

n−1),

(3∗)n =

√1 +

13

I ∗,nτ NRe NDe,

I ∗,nτ = τ ∗,n

rr + 2τ∗,nθθ ,

I ∗,nτ ·τ =

(τ ∗,n

rr)2 + 2

(τ

∗,nθθ

)2,

(4.28)

As extra initial conditions compared to multimode Maxwell we used

D∗rr (0) = L∗

rr (0) = 0,

D∗θθ (0) = L∗

θθ (0) = 0,

(λ∗,n

rr)−1

(0) = 0,

(λ

∗,nθθ

)−1(0) = 0,

(4.29)

since the initial body of fluid is at rest.

4.4.6 Including diffusionIn case of Newtonian material behavior in combination with mass diffusion we rewrite Eq. 4.9into two ordinary differential equations

ddt∗

9 = 9,

ddt∗

9 = 9 =19

[p∗

g − 1 −2

NW e9−

49

NRe9−

3292]

.

(4.30)

These equations are solved in a coupled way and are discretized in time using simple first-order Euler forward, similar to Eq. 4.13.


For p∗g we use Eq. 4.10, discretizing mg(t) in time yields

mn+1g = m0

g + 1t∗n∑

i=0

dmg

dt∗. (4.31)

where dmg/dt∗ is defined by Eq. 4.11.The initial conditions are a combination of the initial conditions given in Eq. 4.14 and

m0g = αβ(t∗)ρB A

43π R3

0, . (4.32)

where β(t∗) is the fraction of the blowing agent contributing to the gas pressure inside thebubble, which is incorporated as a numerical tool to avoid a numerical instability at the startof bubble growth. This factor slows down the increase in gas pressure. Here we assume thefollowing function for β(t∗)

t∗ < aπ : β(t∗) = 12 cos

(t∗a − π

)+ 1

2 ,

t∗ ≥ aπ : β(t∗) = 1,(4.33)

where a is the value determining the increase rate of amount of gas contributing to the gaspressure. Using this β(t∗) results in a smooth increase in the amount of gas present in thebubble contributing to the gas pressure.

4.5 Results: Bubble growth for different constitutive modelsThe results for single bubble growth concerning the in the previous section proposed consti-tutive equations are presented in this section.The dimensionless parameters were presented in Section 4.3. For the Newtonian, Hookeanand Maxwell model, we used the data as given in Table 4.1. The data corresponding toT = 420K are chosen as reference from which we independently varied the fill fraction α,viscosity η0, elastic modulus G0 and external pressure pext . For the multimode Maxwellmodel we choose the temperature equal to 410K and we varied the amount of modes fittedon the master curve created for this temperature (see Table 4.2). In case of the extendedPom-Pom model we adopted the same data set as for the multimode Maxwell model and theintroduced elongational data (see Table 4.3).

4.5.1 NewtonianThe momentum equation describing the single bubble growth for Newtonian material be-havior in dimensionless quantities is rewritten into two ordinary differential equations asdescribed in Section 4.4.1. In comparison to the Maxwell material behavior, the Newtonianmaterial behavior has no elastic effects. Therefore we expect a gradual growth of the singlebubble in time, after which an equilibrium state will be reached.The two ordinary differential equations are solved in a coupled way using a 1t ∗ equal to1 · 10−2. The results have been validated using smaller 1t∗.We study the expansion behavior of the single bubble for different fill fractions α, viscosi-ties η0 and external pressures pext respectively. First of all, we look at the dependence of

52 Chapter 4

0 20 40 60 80 1000

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.1: Radius 9(t∗) for Newtonian material behavior for different temperatures T (in the direc-tion of the arrow respectively 400, 405, 410, 415, 420, 425K ).

the single bubble growth as function of temperature using the input parameters as given inTable 2.2.The results in Figure 4.1 show different behavior for each temperature. The single bubblegrowth per temperature is analyzed and used in Section 3.4.1 to check the values found forthe asymptotical time scales. The equilibrium radii Req achieved at different temperatures arealmost equal, within a deviation of 2%. The small differences are a result from the surfacetension γ and the κ∗

g as function of temperature. To investigate the influences of differentvariables, we chose T = 420K as a reference temperature, from which we started changingsome variables. First, we look at the dependence of the single bubble growth as functionof the fill fraction α. The results in Figure 4.2 show that the time to reach the equilibriumradius 9eq is independent of the fill fraction α, whereas the equilibrium radius 9eq , rangingfrom 7.29 up to 12.47, depends on the fill fraction α. The equilibrium radius 9eq increasesfollowing a logarithmic trend with increasing fill fraction α.

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.2: Radius 9(t∗) for Newtonian material behavior using different fill fractions α (in the direc-tion of the arrow respectively 0.2, 0.4, 0.6, 0.8, 1.0).

Next, we consider the dependence of single bubble growth on the viscosity η0 . As can be


seen in Figure 4.3, the equilibrium radius 9eq is independent of the viscosity, whereas thetime to reach this equilibrium radius 9eq changes with the viscosity η0. The time to reachthe equilibrium radius 9eq increases linearly with increasing viscosity η0.

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.3: Radius 9(t∗) for Newtonian material behavior using different viscosities η0 (in the direc-tion of the arrow from 0.2 · 105 up to 2.0 · 105 Pa · s in steps of 0.2 · 105).

Finally, we vary the external pressure pext and study the dependence on the single bubblegrowth. According to Figure 4.4, the equilibrium radius 9eq and the time to reach this equi-librium radius depend on the external pressure pext . The time to reach the equilibrium radius9eq and the equilibrium radius itself decrease following a logarithmic trend with increasingexternal pressure pext .

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

16

PSfrag replacements

t∗

9

Figure 4.4: Radius 9(t∗) for Newtonian material behavior using different external pressures pext (inthe direction of the arrow from 0.7 · 105 up to 1.3 · 105 N/m2 in steps of 0.1 · 105).

4.5.2 HookeanThe momentum equation describing the single bubble growth for Hookean material behaviorin dimensionless quantities is rewritten into three ordinary differential equations as describedin Section 4.4.2. This material behavior is expected to respond as purely elastic and therefore

54 Chapter 4

we expect an oscillating radius of the single bubble as function of time. The only dependenceon time in this system is a result of inertia. All calculations are performed using a 1t ∗ equalto 4 · 10−3. The results have been validated using smaller 1t∗.For Hookean material behavior we study the growth and oscillation of the single bubble fordifferent fill fractions α, elastic moduli G0 and external pressures pext respectively.The dependence of the single bubble growth as function of temperature using the input para-meters as given in Table 2.2 is shown in Figure 4.5.

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.5: Radius 9(t∗) for Hookean material behavior for different temperatures T (in the directionof the arrow respectively 400, 405, 410, 415, 420, 425K ).

The plot shows a slight influence of the temperature on the oscillation behavior. For each tem-perature, the single bubble growth is analyzed and used in Section 3.4.2 to check the valuesfound for the asymptotical time scales. The influences of different variables, fill fraction α,elastic modulus G0 and external pressure pext , are investigated at T = 420K , the referencetemperature, from which we started to change some variables.First, we investigate the dependence of the single bubble oscillation on the fill fraction α.The results are presented in Figure 4.6. It is shown that the time to reach the equilibriumradius 9eq and the equilibrium radius itself depend on the fill fraction α. The equilibriumradius 9eq and the oscillation time increase following a logarithmic trend with increasing fillfraction α.The results in Figure 4.7 show that the time to reach the equilibrium radius 9eq and theequilibrium radius itself depend on the elastic modulus G0. The equilibrium radius 9eq andthe oscillation time show a logarithmic decrease with increasing elastic modulus G0.The final case for the Hookean material behavior is the dependence of the oscillations onexternal pressure pext . Figure 4.8 shows that the time to reach the equilibrium radius 9eqand the equilibrium radius itself depend on the external pressure pext . The equilibrium radius9eq shows a logarithmic decrease and the oscillation time shows a logarithmic increase withincreasing external pressure pext .

4.5.3 Maxwell model

We consider the momentum equation describing the single bubble growth for Maxwell mate-rial behavior. This equation is rewritten into four ordinary differential equations as described


0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.6: Radius 9(t∗) for Hookean material behavior using different fill fractions α (in the directionof the arrow respectively 0.2, 0.4, 0.6, 0.8, 1.0).

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.7: Radius 9(t∗) for Hookean material behavior using different elastic moduli G0 (in thedirection of the arrow respectively 2.3 · 105 up to 3.1 · 105 N/m2 in steps of 0.2 · 105).

in Section 4.4.3, which are discretized in time and space. Maxwell material behavior is ex-pected to describe a combination of the elastic and viscous growth characteristics for a singlebubble model as earlier described in this section. All calculations are performed using a 1t ∗

equal to 5 · 10−3, but the results have been validated using smaller 1t∗.For Maxwell material behavior we use T = 420K as a reference temperature and study theexpansion of the single bubble for different fill fractions α, elastic moduli G0 and externalpressures pext respectively.First, we calculate the dependence of the fill fraction α on the expansion. The single bubblegrowth per fill fraction value is analyzed and used in Section 3.4.3 to check the values foundfor the asymptotical time scales.In Figure 4.9 is shown that the equilibrium radius depends on the fill fraction α. The equilib-rium radius 9eq increases with increasing fill fraction α. The obtained equilibrium radii equalthe equilibrium radii obtained using Newtonian material behavior and the time scales for theexpansion are of the same order (Figure 4.2). The oscillation times match with Hookeanmaterial behavior (Figure 4.6).

56 Chapter 4

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.8: Radius 9(t∗) for Hookean material behavior using different external pressures pext (in thedirection of the arrow respectively 0.7 · 105 up to 1.4 · 105 N/m2 in steps of 0.1 · 105).

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

16

PSfrag replacements

t∗

9

Figure 4.9: Radius 9(t∗) for Maxwell material behavior using different fill fractions α (in the directionof the arrow 0.2, 0.4, 0.6, 0.8, 1.0).

The results in Figure 4.10 show that the time to reach the equilibrium radius 9eq dependson the elastic modulus G0, whereas the equilibrium radius itself is independent of the elasticmodulus G0. The time to reach the equilibrium radius increases with decreasing elastic mo-dulus G0. The achieved equilibrium radius corresponds to the equilibrium radius found usingNewtonian material behavior. The time before reaching the equilibrium radius increases withincreasing relaxation time (λ0 = η0/G0), which corresponds to the Newtonian case. Thetime scales for oscillation increase with decreasing G0, which corresponds to the Hookeancase (Figure 4.7).In the final case for Maxwell material behavior we consider the expansion as function of theexternal pressure pext . Figure 4.11 shows that the time to reach the equilibrium radius 9eq isindependent of the external pressure pext , whereas the equilibrium radius itself depends onthe external pressure pext . The equilibrium radius increases with decreasing external pressurepext . The obtained equilibrium radii and the time scales for expansion equal Newtonianmaterial behavior. The time scales for oscillation decrease correspond to the order of thetime scales for oscillation as found for Hookean material behavior.


0 10 20 30 40 50 60 70 800

4

8

12

16

20

PSfrag replacements

t∗

9

Figure 4.10: Radius 9(t∗) for Maxwell material behavior using different elastic moduli G0 (in thedirection of the arrow 6 · 105 down to 105 N/m2 with steps of 105).

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

16

PSfrag replacements

t∗

9

Figure 4.11: Radius 9(t∗) for Maxwell material behavior using different external pressures pext (inthe direction of the arrow 1.1 · 105 down to 7 · 104 N/m2 in steps of 104).

From the plots concerning Hookean material behavior, we can conclude that the time scalesfor oscillation are of the order 10. From the plots showing Newtonian material behavior,the time scales are of the order 102. Comparing these time scales to the Maxwell materialbehavior, we can see that the oscillations are indeed of the order 10 and the expansion is ofthe order 102.

4.5.4 Maxwell model: From pure elastic to NewtonianThe Maxwell model is a combination of the Newtonian and Hookean model. These twolimiting cases can be obtained by varying the relaxation time λ0. Increasing or decreasingthe relaxation time λ0 in case of Maxwell material model reaches Hookean (λ0 → ∞) orNewtonian (λ0 → 0) material model respectively. To reach Hookean material behavior, theviscosity η0 is increased, only influencing NDe and NRe , see Figure 4.12.To reach the Newtonian material behavior, the elastic modulus G0 is increased, only influ-encing NDe, see Figure 4.13.

58 Chapter 4

0 10 20 30 40 50 600

2

4

6

8

10

12

14

16

PSfrag replacements

t∗

9

Figure 4.12: Radius 9(t∗), changing from Maxwell to Hookean material behavior using different vis-cosities η0 (in the direction of the arrow 1.05 ·105, 2.05 ·105, 5.05 ·105, 2.05 ·106 Pa · s),T = 420K .

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

16

PSfrag replacements

t∗

9

Figure 4.13: Radius 9(t∗), changing from Maxwell to Newtonian material behavior using differentelastic moduli G0 (in the direction of the arrow 2.72 · 105, 5.50 · 105, 1.50 · 106, 5.50 ·106 N/m2), T = 420K .

For both situations eliminating either the viscous or the elastic effects results in a situationcorresponding to the original cases applying the pure Hookean or Newtonian material model.

4.5.5 Multimode Maxwell model

The multimode approximation is often necessary for a realistic description of the viscoelasticcontributions. We will show the bubble growth as function of time for different number ofmodes. All calculations are performed using time step 1t∗ equal to 6 · 10−3, but the resultshave been validated using smaller 1t∗.Figure 4.14 shows the single bubble growth for different number of modes. For one mode,the growth looks for short times similar to Hookean oscillation. Upon increasing the numberof modes, the bubble radius as function of time tends to Maxwell model growth as can beseen in the previous section.


0 20 40 60 80 1000

5

10

15

20

25

30

PSfrag replacements

t∗

9

Figure 4.14: Radius 9(t∗) for the multimode Maxwell model using different number of modes (in thedirection of the arrow 1, 2, 4 and 6) at T = 410K .

4.5.6 Extended Pom-Pom model

To capture the non-linear behavior in both elongation and shear, we used the extended Pom-Pom (XPP) model (Verbeeten et al., 2001). We show the bubble growth as function of timefor different q, the number of arms at the end of a backbone, as presented in Table 4.3.Increasing the value of q as given from q1 to q2 increases the strain hardening effect. Bothcalculations are performed using the relaxation spectrum consisting of 6 modes and usingtime step 1t∗ equal to 4 · 10−3, but the results have also been validated using smaller 1t∗

and are shown in Figure 4.15.

0 10 20 30 40 50 60 70 800

5

10

15

20

25

q2

q1

PSfrag replacements

t∗

9

Figure 4.15: Radius 9(t∗) for the extended Pom-Pom model using different q as presented in Table 4.3at T = 410K

The plots as shown in Figure 4.15 are comparable to plots for the multimode Maxwell model.Going from q1 to q2 increasing the strain hardening effect shows a minor decrease in theperiod and amplitude of oscillation.

60 Chapter 4

4.6 Influence of diffusionThe theory concerning the diffusion of the blowing agent is independent of the rheologicalbehavior of the shell material. To show the importance of the diffusion on the actual growthwe chose the momentum equation as derived for the Newtonian model as is shown in Sec-tion 4.2.2.Since we introduced a new factor β (Eq. 4.33) as the fraction of blowing agent contributing tothe gas pressure inside the bubble, we plotted β as function t∗ for different a in Figure 4.16,where a determines the rate of increase of β. Figure 4.16 shows that the rate of increase of β

decreases with increasing a.

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PSfrag replacements

t∗

β

Figure 4.16: β(t∗) with varying a (in the direction of the arrow respectively 6, 8, 10, 12, 14).

To show the influence of the factor β on the bubble growth, we compared the results asgenerated without diffusion for a = 0 (then β = 1) and a = 10. From Figure 4.17 we cansee that the bubble growth process is only slowed down by the incorporation of Eq. 4.33,whereas the trend stays equal.

0 20 40 60 80 1000

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

a = 0

a = 10

Figure 4.17: Radius 9(t∗) with D equal to zero and a = 0 and a = 10 at 420K .

In this section, we study the influence of the value of the diffusion coefficient D on thegrowth of the bubble. To do so, we consider the expansion at 420K using a = 10 and vary


D independently of all variables. The radius as function of time for different D is given inFigure 4.18, showing that the diffusion has an enormous impact on the growth of the bubble.For large enough D, it forces the bubble to collapse after the maximum radius has beenachieved.

0 20 40 60 80 100 120 140 160 1800

2

4

6

8

10

12

14

PSfrag replacements

t∗

9

Figure 4.18: Radius 9(t∗) for Newtonian material behavior at 420K with kh = 10−7cm3/(cm3 · Pa)

and varying diffusion coefficient D (in the direction of the arrow respectively 1 · 10−9, 1 ·10−5, 1 · 10−4, 1 · 10−3, 1 · 10−2m2/s).

We can also plot the corresponding mg(t∗)/ml , the actual mass of gas present in the bubbledivided by the initial liquid mass of blowing agent in the bubble, as function of time t ∗, seeFigure 4.19. The increase in mg(t∗)/ml is a result of the increasing β, whereas the decreaseis a result of the diffusion. For increasing diffusion coefficients, we see that the maximum ofmg(t∗)/ml decreases and thus lowering the maximum possible expansion.

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PSfrag replacements

t∗

mg(

t∗ )/

ml

Figure 4.19: Relative mass of gas mg(t∗)/ml (the actual mass of gas present in the bubble divided bythe initial liquid mass of blowing agent in the bubble) as function of time t∗ at 420K withkh = 10−7cm3/(cm3 · Pa) and varying diffusion coefficient D (in the direction of thearrow respectively 1 · 10−8, 1 · 10−5, 1 · 10−4, 1 · 10−3, 1 · 10−2m2/s).

From experiments as presented in Chapter 5 we know that the diffusion plays a major role inthe expansion of WEPS. From Figure 4.18 and 4.19 we can conclude that the starting pointfor our diffusion coefficient D equal to 10−9m2/s does not cause the loss of blowing agent as

62 Chapter 4

experimentally observed. Increasing D essentially, shows the expected collapse of the singlebubble. Whereas we have to note that the loss of blowing agent is also strongly dependenton the solubility factor kh (permeability = solubility × diffusivity), which is for consistencykept constant in this chapter.Thus, the diffusion coefficient of water through polystyrene above 100oC (the glass transi-tion temperature of polystyrene and the boiling point of water) has indeed to be higher thenpredicted by the extrapolation based on experimental data achieved below 100oC (see Ap-pendix A), which is already mentioned in Section 2.6.2.Besides the diffusion of blowing agent, heat transfer has to be taken into account, increasingthe effect of diffusion by slowing down the bubble growth. The used β could be coupled tothe heat transfer equation. As can be seen below, the β has a major influence on the growthof the single bubble.To check the dependence of a on the bubble growth we plot in Figure 4.20 the radius 9(t ∗) asfunction of time t∗ for D equal to the theoretical values 10−2m2/s, kh equal to 10−7cm3/(cm3·Pa) and different a. As can be seen in Figure 4.20, increasing a shows a maximum in theachieved radius 9(t∗). This is a result of the pressure dependence of the diffusion.

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

10

PSfrag replacements

t∗

9

Figure 4.20: Radius 9(t∗) for Newtonian material behavior at 420K with constant diffusion coefficientD = 10−2m2/s, kh = 10−7cm3/(cm3 · Pa) and different a (in the direction of the arrowrespectively 0.1, 1, 3, 5, 7, 10, 13, 20, 30, 40, 50).

The corresponding relative mass of gas is given in Figure 4.21. From Figure 4.21 we can seethat a rapid increase in mg is combined with a rapid decrease.

4.7 ConclusionsIn this chapter we have introduced different numerical schemes to solve the momentum equa-tion in combination with a variety of constitutive equations. Moreover, the influence of thediffusion was successfully incorporated, showing collapse of the single bubble.We used our numerical data to validate the asymptotical analysis as presented in Chapter 3for the Maxwell model and both extremes, the Newtonian and the Hookean model. Thisshowed a good agreement and for this reason we can conclude that our numerical analysisas well as our asymptotical analysis for the three mentioned constitutive equations provideaccurate results. Another calculation that improved our confidence concerning the numerical


0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

PSfrag replacements

t∗

mg/

ml

Figure 4.21: Relative mass of gas mg/ml (the actual mass of gas present in the bubble divided by theinitial liquid mass of blowing agent in the bubble) as function of time t∗ at 420K withconstant diffusion coefficient D = 10−2m2/s, kh = 10−7cm3/(cm3 · Pa) and differenta (in the direction of the arrow respectively 0.1, 1, 3, 5, 7, 10, 13, 20, 30, 40, 50).

analysis for the three above mentioned constitutive equations is the following: We end up atthe numerically calculated radii as function of time for the Hookean and Newtonian model bystarting with the Maxwell constitutive model in combination with increasing or decreasingthe relaxation time respectively. With this in mind we changed several input parametersindependently to find their influence on the bubble growth.A multimode approximation is often necessary for a realistic description of the viscoelasticcontributions. Varying the number of modes used for fitting the data as presented in Chap-ter 2 results in different time scales for expansion. For a single mode we can conclude thatthe expansion considering short times shows a behavior as exhibited by using the Hookeanmodel, whereas for more modes, this behavior changes more and more to Maxwell modelgrowth of the single bubble.To capture the non-linear behavior in both elongation and shear, we used the extended Pom-Pom (XPP) model (Verbeeten et al., 2001), which is only used to investigate the influence ofstrain hardening on our bubble growth. The effect of strain hardening appears to be moderatefor a wide range of parameters (see Table 4.3).The numerically calculated radius as function of time using all above mentioned constitutivemodels shows oscillations except for the Newtonian model. In real-life, expandable poly-styrene (EPS) beads do not oscillate upon expansion. The oscillations shown are expectedto be a result of the combination of the initially applied gas pressure and the bubble shellmaterial behavior described by the different constitutive equations. Applying a heat transferequation, herewith gradually increasing the gas pressure inside the bubble, is expected to ex-clude at least some of the oscillations. However, the investigated influences of the differentparameters on the bubble growth will remain valid.Another investigated phenomena is the diffusion of blowing agent to the exterior of the sys-tem. From experimental results (see Chapter 5) we can see that diffusion, especially for ourchosen model system, has a major effect on the bubble growth. Using the value for the diffu-sion coefficient as proposed in Section 2.6.2, is of hardly any influence on the bubble growth.This suggested using substantially larger D than predicted by the extrapolation based on ex-perimental data achieved below 100oC (see Appendix A). This is in agreement with the in

64 Chapter 4

literature mentioned D for small molecules through a polymer above the glass transition tem-perature of the polymer. Applying a substantially larger D resulted in growth of the bubble,subsequently followed by a collapse.

Chapter 5

Suspension polymerization andexpansion of WEPS

5.1 IntroductionIn the previous chapters, the asymptotical calculations using the single bubble model werevalidated using the numerical calculations and vice versa. This chapter provides the possibi-lity of comparing our computational data with experimental results.First of all, we synthesize different grades of water expandable polystyrene (WEPS). Creve-coeur et al. (1999a) developed a two step WEPS-recipe. In the first step, water is emulsifiedby means of a suitable emulsifier in the monomer styrene. Polymerization using a radicalinitiator results in a pre-polymerized styrene/PS mixture possessing a high viscosity, fix-ating the emulsified water droplets. In the second step, this so-called inverse emulsion issuspended in water containing a suitable suspension agent and the polymerization is com-pleted. Using this recipe, we prepared WEPS beads with varying water content and molarmass. The molar mass is related to the melt strength of the polymer matrix, changing theexpansion behavior of the bead. Another possibility to influence the melt strength is thehomogeneous dissolution of end-capped poly(2, 6-dimethyl-1, 4-phenylene ether) (PPE) instyrene prior to the first step. A new challenge is to use the applied WEPS-route for the pro-duction of a water expandable homogeneous blend of polystyrene and poly(2, 6-dimethyl-1, 4-phenylene ether) (PPE), referred to as WE(PS/PPE). The commercial expandable blendof PS/PPE (Dythermr) still contains pentane-isomers as the blowing agent (developed byNelissen and Zijderveld (1990)).In order to investigate the expansion characteristics of the synthesized grades in a fast andreliable way, a laboratory scale expansion setup has been developed and used.Finally, in Chapter 6, the obtained expansion characteristics are compared with the computa-tional data as presented in Chapter 4.

5.1.1 Suspension polymerizationThe standard method for manufacturing EPS beads is a batch-wise suspension polymerizationprocess. A standard textbook recipe of a free-radical polymerization in a suspension processstarts by dispersing the liquid monomer styrene in a continuous aqueous phase, stabilized

66 Chapter 5

by stirring and a suspension stabilizer (Grulke, 1990). The mixture is heated to a suitablepolymerization temperature. After a conversion of about 65 − 70% the blowing agent, avolatile organic compound (VOC), is added to the vessel (approximately 5 − 8wt .%). After100% conversion the obtained spherical polystyrene beads are immediately commerciallyapplicable.In case of WEPS, only water as a blowing agent, the EPS process has to be changed, sincewater is immiscible with polystyrene. Crevecoeur et al. (1999a) developed a bulk polyme-rization step prior to the free-radical polymerization in a suspension process. In this step aphysical surfactant is added to disperse the blowing agent water in styrene (yielding a w/oemulsion). After pre-polymerization, the viscosity should be sufficient to fixate the emulsi-fied blowing agent. If the viscosity is insufficient the system will transfer to a single emulsioninstead of the desired inverse emulsion in suspension (w/o/w system).After the pre-polymerized mass possesses a desired viscosity (proportional to conversion),the reaction mixture is suspended in water in the presence of a suitable suspension stabilizer.The degree of conversion in the mixture influences the break up and coalescence mechanismsand thus the final bead size. It has been reported that a slight transfer of the physical surfac-tant from the reaction mixture to the suspension medium may not be completely excluded,influencing the suspension stability. At the same time, the entrapment of suspension water inthe PS beads may not be ignored, resulting in a higher water content in the beads than maybe expected from the initial amount of emulsified water.

5.1.2 Expansion of WEPSThe expansion process of WEPS and derivatives differs fundamentally from EPS. The mainreasons for this different behavior are:

1. Water is immiscible in PS and PPE, therefore the blowing agent of WEPS has to bedispersed in the polymer matrix as discrete droplets. The size and the dispersion of thewater droplets is expected to determine the foam characteristics and the initiation ofcell growth, every droplet is expected to be an individual nucleus.

2. A trivial fact: the molar mass of water is low compared to pentane (18 vs. 72g/mol).As a result, WEPS may theoretically use 25wt .% of the blowing agent as used for EPSto achieve a comparable foam density.

3. In the case of EPS, pentane has a plasticizing effect on polystyrene, reducing the glasstransition temperature Tg (to approximately 70oC). For this reason and the low boilingpoint of the blowing agent Tbp ≈ 32oC , the commercial expansion process, usingsaturated steam as heating medium, is applicable. For WEPS, the Tg of the bead is100oC (no plasticizing effect of water on PS), which is equal to the boiling point ofthe blowing agent and to the temperature of saturated steam. For this reason, thisheating medium cannot be used for WEPS. For WE(PS/PPE), the Tg of the beads arehigher than 100oC (maximum 30wt .% PPE results in Tg ≈ 130oC), which makessaturated steam also impossible as a heating medium for expansion. An applicableheating medium for both systems would be hot air or superheated steam at temperaturessubstantially above 100oC .

4. The diffusion coefficient of water through PS or PS/PPE at elevated temperatures(above the Tg) is higher compared to the diffusion coefficient for pentane through PS

Suspension polymerization and expansion of WEPS 67

(Vrentas and Duda, 1976). Applying hot air as the heating medium results in a largeconcentration difference of water inside and outside the bead. This difference, thedriving force for diffusion, in combination with a temperature above the Tg leads toa major loss of blowing agent without foaming. Therefore, superheated steam is sug-gested, resulting in a higher concentration of blowing agent outside the bead and thusin a diminished loss of blowing agent.

5.2 Experimental

5.2.1 Suspension polymerization

Materials

Styrene monomer was used without distillation from the inhibitor tert-butylcatechol (11 −17ppm) and was supplied by Caldic. Radical initiators tert-butyl-2-ethyl-hexylperoxicarbo-naat (t-BEHC) with t1/2 = 110 minutes at 120oC , dibenzoylperoxide (DBPO, active content75%, water 25%) with t1/2 = 75 minutes at 90oC , the surfactant SURF, potassium persul-fate (PPS), the chain transfer agent 2,4-diphenyl-4-methyl-1-pentene (DMP) and the standardsuspension stabilizer were supplied by NOVA Chemicals, Breda and were used as received.End-capped PPE was supplied by GE Plastics, Bergen op Zoom, PPE-800.

Polymerization recipes

Two main formulations were developed for the preparation of WEPS beads. For both formu-lations the surfactant, SURF (NOVA Chemicals Breda, 2002), has been used to stabilize thedispersed water in styrene during the pre-polymerization. The overall production setup con-sists of a 5 liter pre-polymerization reactor and a 10 liter suspension polymerization reactor,see Figure 5.1.The used ratio of reaction mixture/suspension medium was 1. In practise we used overall 4liters organic phase (styrene including PPE in case of Formulation P) and 4 liters water.The first formulation (R) represents the standard recipe for WEPS. The second one, Formu-lation P is equal to Formulation R but prior to the pre-polymerization step PPE is dissolvedhomogeneously in the styrene monomer. Weight percentages on total matrix (PS/PPE) basis(wt .%M), on styrene basis (wt .%S) and on water basis (wt .%W ) are used.

Formulation RThe surfactant (SURF, 0.5wt .%M), the initiator (DBPO, 0.22wt .%S) and the blowing agentwater (containing sodium chloride) were added to styrene and the mixture was heated to90oC and polymerized in bulk for 160 − 165 minutes while stirring at 400 r.p.m. (onlinedetermination of the conversion was not possible due to the reactor setup). Subsequently, theviscous reaction mixture was suspended in water in the suspension vessel. The suspensionstabilizer was dissolved in the suspension water and an extra amount of initiator (t-BEHCand DBPO) was added to obtain full conversion. When applicable, DMP was added to thesuspension vessel. The polymerization was continued for 120 minutes at 93oC while stirringat 500 − 600 r.p.m. Hereafter, the temperature was slowly increased to 120oC and the poly-merization was continued for another 60 minutes. Finally, the mixture was cooled to roomtemperature and the spherical beads were separated and washed with water.

68 Chapter 5

o i l b a t h

T

T I C

T

P

o i l b a t h

T I C

P

F

w a t e rs u s p e n s i o n s t a b i l i s e r

n i t r o g e n

w a s h w a t e r

s o l i d P S b e a d s

f i l t e r

s t y r e n es u r f a c t a n ti n i t i a t o rw a t e r

p r e - p o l y m e r i s a t i o nr e a c t o r

s u s p e n s i o n p o l y m e r i s a t i o nr e a c t o r

Figure 5.1: The used reactor setup at NOVA Chemicals, Breda, consisting of a pre-polymerization (5l)and a suspension polymerization reactor (10l), with TIC: temperature indicated controller,T: temperature indicator, P: pressure indicator, F: flow indicator.

Table 5.1: The compounds for the pre- and suspension polymerization formulations in parts by weightfor Formulation R (WEPS).

pre- suspension polymerizationNaCl PPS t-BEHC DBPO DMP

R-grade [wt .%W ] [10−3wt .%W ] [10−2wt .%S] [10−1wt .%S] [wt .%S]1 0.0 0.9 5.5 3.0 -2 0.1 0.9 5.5 3.0 -3 0.3 1.5 5.5 3.0 -4 0.1 1.5 7.5 1.5 -5 0.1 1.5 5.5 3.0 0.07

Formulation PAs Formulation R, but prior to the emulsification of water, PPE was dissolved homogeneouslyin styrene. In the pre-polymerization step we used 0.1wt .%W NaCl, in the suspension poly-merization step we used 1.5·10−3wt .%W PPS, 5.5·10−2wt .%S t-BEHC and 3.0·10−1wt .%SDBPO. Furthermore, the pre-polymerization time was reciprocally reduced with the PPEconcentration. Several grades were synthesized using the PPE fractions P1: 5wt .%M , P2:10wt .%M , P3: 20wt .%M and P4: 30wt .%M .


Characterization techniques

Differential scanning calorimetry (DSC)The glass transition temperature Tg of PS/PPE samples was determined using a Perkin ElmerDSC Pyris 1. The heating rate was 20K/min and for each sample two heating runs from 50to 240oC were recorded. Indium was used for temperature and heat of fusion calibration.

Gel permeation chromatography (GPC)Gel permeation chromatography was used to determine the number average molar mass (Mn),the weight average molar mass (Mw), and the molar mass distribution (Mw/Mn) of the PS.The samples were measured on a Waters modular GPC system consisting of a pump M510,auto sampler 717, detector 996 (UV at 254 nm), detector 410 (refractometer) and PolymerLabs Columns. Temperature of column and detector were kept at 40oC . Eluent was THF(p.a., stabilized) at 0.8 ml/minute with an injection volume of 50ml and a sample concentra-tion of 2 mg/ml. The columns were calibrated with Toyo Soda polystyrene standards in therange of Mw = 3.84 · 106 − 946g/mol.

Thermogravimetric analysis (TGA)In order to determine the water content of the WEPS beads, the weight loss of a sieve frac-tion of 1.7 − 2.0mm beads was measured using Perkin Elmer TGA6 under a nitrogen flow of20ml/min. The temperature program started at 30oC at a heating rate of 40K/min to 140oCfollowed by an isothermal period of 60min.

Scanning electron microscopy (SEM)SEM was performed on all samples using a SEM hivac at 2kV to visualize the water dropletsize and dispersion. Compact beads were microtomed to obtain smooth surfaces, whereasfoamed beads were fractured at liquid nitrogen temperature and subsequently coated with15nm chroom.

5.2.2 ExpansionLabscale expansion setup

In Figure 5.2 the used single bead labscale expansion setup is drawn.

TT

PSfrag replacements

digital camera attached to microscopethermocouple

micro-bath containing silicone oil

expandable bead attached to a needle

silicone oil

Figure 5.2: Schematic representation of the labscale apparatus for a single bead expansion experiment.

This expander is modified after the by Fen-Chong et al. (1999) proposed expansion setup.The double walled micro-bath (V ≈ 10ml) contains silicone oil as the heating medium. Afterfixation to a needle, one bead at the time was added to this bath. A microscope with a digital

70 Chapter 5

videorecorder was placed above the bath. This allowed us to record online the diameter ofthe bead during the expansion process (see Figure 5.3 and 5.4). The average relative volumeof the bead (volume of the expanded bead divided by the volume of the initial compact bead)as function of time was calculated from three experiments of different beads from the samegrade.

PSfrag replacements

t = 0s t = 10s t = 20s

t = 30s t = 40s t = 50s

t = 60s t = 70s t = 80s

Figure 5.3: The expansion of a EPS bead, initial bead size: 1.3mm, at Tbath = 105oC (the dropletscoming out of the bead are release of blowing agent).

PSfrag replacements

t = 0s t = 2s t = 4s

t = 6s t = 8s t = 10s

t = 12s t = 14s t = 16s

Figure 5.4: The expansion of a WEPS bead, initial bead size: 1.4mm, at Tbath = 150oC (the dropletscoming out of the bead are release of blowing agent).


5.3 Results and discussion

5.3.1 Suspension polymerization

WEPS according to Formulation R

The starting point of our experimental research was the recipe as described by Crevecoeuret al. (1999a). In this procedure the surfactant AOT (bis(2-ethylhexyl)sulfosuccinate) wasused together with hydrophilic copolymers as the suspension stabilizer. Due to the unavail-ability of the hydrophilic copolymers we changed to a standard suspension stabilizer. Thecombination of AOT with this stabilizer resulted in frequent suspension failures. The reasonfor this is that a fraction of AOT is transferred, due to an osmotic effect, from the reaction mix-ture to the suspension system, decreasing the suspension stability. To avoid this undesirableeffect, the use of the industrial surfactant SURF was proposed by NOVA Chemicals Breda(2002). By using the combination of surfactant SURF and the standard suspension stabilizeras proposed, a stable and reproducible suspension has been achieved. At the mean time, dif-ferences in water droplet size and dispersion in the beads can be seen, as clearly depicted inFigure 5.5.

PSfrag replacements

200µm200µm A B

Figure 5.5: SEM micrograph of compact WEPS bead, A: using AOT as the surfactant. B: with SURFas the surfactant.

From the micrographs, we can conclude that the droplet size by using SURF is essentiallyincreased; in other words the surfactant SURF is less powerful and is therefore expected notto be able to disturb the suspension as has been experienced by using AOT.Several WEPS-grades with different amounts of water (blowing agent), but with comparablemolar mass, droplet size and droplet dispersion (R1, R2 and R3) have been prepared. This isexperimentally achieved by adding NaCl to the emulsified water. NaCl induces an osmoticeffect from the suspension water to the emulsified water, resulting in an overall higher watercontent with increasing NaCl concentration. This is found to be partly true, as listed inTable 5.2 (R1, R2 versus R3). The water droplet dispersion changes for the three samplesR1, R2 and R3 as is visualized in Figure 5.6. From these observations, we can conclude thatthe concentration of NaCl shows an optimum concerning the amount of blowing agent andshows an optimum in the droplet size and dispersion. In this study R3 shows an undesirabledroplet size and dispersion, since the droplets are extremely inhomogeneous in size and indispersion.

72 Chapter 5

PSfrag replacements

200µm 200µm200µmR1 R2 R3

Figure 5.6: SEM micrograph of compact WEPS beads, R1: 2.1wt.% water (no NaCl), R2: 3.3wt.%water (0.1wt.% NaCl), R3: 2.8wt.% water (0.3wt.% NaCl).

To investigate the influence of the melt strength of the polymer matrix on the expansionbehavior, we prepared grades with different molar mass. A decrease in initiator concentrationresults in an increase of the molar mass (grade R4) and the use of the chain transfer agentDMP results in a decrease of the molar mass (grade R5), where the molar mass is proportionalto the melt strength.

PSfrag replacements

200µm200µm R4 R5

Figure 5.7: SEM micrograph compact WEPS beads, R4: Mw = 237kg/mol , 3.7wt.% water, R5:Mw = 163kg/mol , 3.6wt.% water.

Table 5.2: The incorporated water content, the weight average molar mass (Mw) and the molar massdistribution (Mw/Mn ) of WEPS-grades after Formulation R.

Incorporated Mw Mw/MnR-grade water [wt .%] [kg/mol] [-]

1 2.1 167 2.32 3.3 173 2.13 2.8 177 2.14 3.7 237 2.45 3.6 163 2.3

From this study we can conclude that, taking into account the reproducibility, the suspension


stability and the water droplet size and dispersion, the obtained WEPS grades R1 to R5are promising for obtaining WEPS-foam. Grade R4 is expected to have an increased meltstrength, whereas grade R5 is expected to have a slightly decreased melt strength. A betterway of achieving an increased melt strength is by adding a melt strength ’thickener’. In thesubsequent section PPE is used for this reason.

WE(PS/PPE) according to Formulation P

For WEPS we increased the melt strength of the polymer matrix by decreasing the initiatorconcentration and the addition of a chain transfer agent. Another possibility to increase themelt strength of the polymer matrix (and so fixating the water droplets) is the addition ofend-capped PPE, which resulted in the P-grade. PPE is a suitable polymer for styrene forthree reasons:

1. PPE can be dissolved homogeneously in styrene at ambient temperatures.

2. After polymerization of styrene a miscible PS/PPE blend, for all compositions, is ob-tained (Nelissen et al., 1990).

3. PPE possesses the possibility to act as a melt strength ’thickener’ and therefore the pre-polymerization step (conversion of styrene) can be reduced substantially (the higher thePPE concentrations, the higher the reduction).

From the grades produced by Formulation R, grade R2 shows the best droplet dispersion(Figure 5.6) and is therefore used as a starting point for Formulation P. The main differencebetween the formulation for grade R2 and Formulation P in general is the addition of PPE.For the first time, we were able to prepare WE(PS/PPE) beads successfully with differentamounts of PPE. The amount of incorporated water and the glass transition temperature Tg ofthe polymer matrix are given in Table 5.3. In Figure 5.8, the water droplet size and dispersionare shown for the obtained WE(PS/PPE) beads.

Table 5.3: The water content and the glass transition temperature of compact WE(PS/PPE) beads, afterFormulation P.

Incorporated TgP-grade water [wt .%] [oC]

1 9.4 1042 8.1 1093 6.2 1184 1.2 128

The amount of blowing agent in the final beads, at least for P1, P2 and P3 is higher thanthe amount of emulsified water during the pre-polymerization and is higher than in the caseof WEPS. This effect is a result of the hydrophilic behavior of PPE. Striking is the inverserelation between PPE concentration and the incorporated water content, which is expected tobe a result of the ternary system, PS/PPE/water, but further investigation is necessary.

74 Chapter 5

PSfrag replacements

200µm 200µm

200µm200µm

P1 P2

P3 P4

Figure 5.8: SEM micrograph of compact WE(PS/PPE) bead, P1: Tg = 104oC , 9.4wt.% water, P2:Tg = 109oC , 8.1wt.% water, P3: Tg = 118oC , 6.2wt.% water, P4: Tg = 128oC ,1.2wt.% water.

5.3.2 Expansion

The main purpose of the synthesis according to the Formulations R and P was to accuratelytrack, in a fast and reproducible way, the relative volume of the bead (volume of the ex-panded bead divided by the volume of the initial compact bead) as function of expansiontime (foaming behavior) for all synthesized samples. In Chapter 6, the obtained expansioncharacteristics are compared with our numerical analysis. For this purpose, the developedlabscale expander (see Figure 5.2) was used. We realize that this expander is economicallynot feasible, but for usage on labscale, it is a perfect, fast and reproducible tool. From theachieved relative volumes as function of time, we can determine different expansion rates,maximum expansion values and collapse rates.

Expansion behavior of commercial EPS in oil

In order to be able to compare the expansion characteristics of the different WEPS beadswith commercially applied EPS, the expansion of the latter one was examined using oursingle bead expander, resulting in Figure 5.9.From these data we can conclude that after a delay of a few seconds, the bead grows conti-nuously, achieving a maximum expansion. When the expanded bead is not cooled under theglass transition temperature, and the (slow) diffusion of the blowing agent to the exterior of


0 50 100 150 2000

2

4

6

8

10

12

PSfrag replacements

t[s]

V/

V 0[−

]

Figure 5.9: The relative volume V/V0(t) for commercial EPS, containing 7wt.% pentane, at Tbath =105oC .

the bead will continu, collapse will occur, and thus a decrease in V/V0 will be the result.

Expansion characteristics of WEPS

For the water expandable grades we are trying to visualize the differences in expansion cha-racteristics. Less important is to find the optimum expansion temperature and the maximumexpansion. After several foaming experiments at different temperatures, we decided to con-tinue using Tbath = 150oC as the foaming temperature in our expander.We investigated all R-grades in our labscale expander by online monitoring the expansionprocess. The results for R1, R2 and R3 (Mw ≈ 170kg/mol) are shown in Figure 5.10.

0 5 10 150.5

1.5

2.5

3.5

4.5

I II III

PSfrag replacements

t[s]

V/

V 0[−

]

Figure 5.10: The relative volume V/V0(t) of WEPS at Tbath = 150oC , R1: ◦, 2.1wt.% water, R2: �,3.3wt.% water, R3: G, 2.8wt.% water.

All three samples posses a different expansion behavior compared to EPS. The main diffe-rence is the prominent presence of the collapse of the bead as shown in Figure 5.10. Duringthe expansion of WEPS three pronounced stages can be distinguished: (I) induction, (II) pro-cessing window, including expansion up to a maximum and initiation of collapse and (III)collapse. Whereas in the case of EPS (Figure 5.9), Stage III is not observed for the used

76 Chapter 5

time scale (tmax ≈ 200s). This difference is a result of the fast diffusion of water/steam(Figure 5.4) in contrast with the slow diffusion of pentane (Figure 5.3) from the bead to itsexterior. In case of WEPS, the diffusion already starts in Stage I, where the vapor pressureinside the bead is not enough for expansion, but the temperature of the polystyrene matrix isalready above the glass transition temperature of polystyrene. For this reason no expansionoccurs in Stage I.Considering the differences for the R samples as shown in Figure 5.10 we can conclude thata higher concentration of water does not necessary result in a higher maximum expansion.The water droplet dispersion and size (R1 versus R3) seem to be more dominant, see SEMmicrographs in Figure 5.6. The differences of the relative volume as function of time betweenR1 and R2 is within the experimental accuracy of the V measurements, but is not in conflictwith the previous conclusion. From these expansion results, the main concern is the smallprocessing window (Stage II) during foaming and the sharp decrease of V/V0 as functionof time in Stage III. We expect that an increased melt strength delays the collapse processand thus increases Stage II. For this reason we synthesized grades with different molar mass.Grade R4 possesses a higher molar mass and grade R5 possesses a lower molar mass, withthe molar mass proportional to the melt strength. The influence of the melt strength on theactual expansion is shown in Figure 5.11.

0 5 10 150.5

1.5

2.5

3.5

4.5 I II III

PSfrag replacements

t[s]

V/

V 0[−

]

Figure 5.11: The relative volume V/V0(t) for different melt strength of WEPS at Tbath = 150oC , forgrade R4: �, Mw = 237kg/mol , R5: ◦, 163kg/mol .

Both samples (R4 and R5) posses different expansion behavior, which is particularly inter-esting since both samples contain approximately equal amounts of blowing agent (3.7 ver-sus 3.6wt .%) with comparable water droplet dispersion (see Figure 5.7), but different meltstrength. We can conclude that R4 reaches, as expected, its maximum expansion later thanR5 and that the slope of collapse is higher for R5 than for R4. Thus an increased melt strengthresults in an increased processing window (Stage II) as shown in Figure 5.11.From this investigation we can conclude that by increasing the melt strength, the processingwindow broadens, but unfortunately, collapse cannot be prohibited.

Expansion characteristics of WE(PS/PPE)

Another route to increase the melt strength is to add PPE to styrene, resulting after polymeri-zation in miscible PS/PPE blends containing water as the blowing agent (WE(PS/PPE)). The


WE(PS/PPE) beads possess a comparable gap, Tg > Tbp, for expansion as also present forEPS, see Table 5.4.

Table 5.4: The boiling point Tbp of the blowing agents and the glass transition temperatures Tg of thematrix for different expandable systems.

Tg [oC] Tbp [oC]EPS ∼ 70 28 − 36WEPS ∼ 100 ∼ 100WE(PS/30wt .%PPE) ∼ 130 ∼ 100

For the WE(PS/PPE) grades, the same labscale expander was used as applied for WEPS atthe same expansion temperature, Tbath = 150oC . In Figure 5.12, we see for grade P3 a largermaximum expansion than found for all other P- and R-grades. This could be a result of anoptimum expansion temperature for the glass transition temperature Tg of P3 in combinationwith Tbp.

0 2 4 6 8 10 12 14 16 180.5

1.5

2.5

3.5

4.5

5.5

6.5

I II III

PSfrag replacements

t[s]

V/

V 0[−

]

Figure 5.12: The relative volume V/V0(t) for WE(PS/PPE) at Tbath = 150oC , P1: ◦, 5wt.% PPE,P2: �, 10wt.% PPE, P3: G, 20wt.% PPE, P4: ∗, 30wt.% PPE.

Comparing WE(PS/PPE) with WEPS, we can distinguish a longer induction period (StageI), an increased processing window (Stage II) and less collapse (Stage III), as a result of theincreased melt strength.At the maximum expansion, we extracted some beads from the single bead expander andquenched the beads at liquid nitrogen temperature. Subsequently, SEM analysis was per-formed. Figure 5.13 visualizes that the achieved foam structures are not regular, which couldbe a result of the large droplet size in the compact WE(PS/PPE) beads as is clearly visualizedin Figure 5.8.A magnification (2.5×) of SEM micrograph C (Figure 5.13) shows a foam structure as ismore comparable with the well-known EPS or WEPS foam structure (Figure 5.14).In the presented study we successfully applied the WEPS process (polymerization and foam-ing) for the PS/PPE system.

78 Chapter 5

PSfrag replacements

500µm 500µm

500µm500µm

P1 P2

P3 P4

Figure 5.13: SEM micrographs of WE(PS/PPE) foam, expanded at 150oC , P1: Tg = 104oC , 9.4wt.%water, P2: Tg = 109oC , 8.1wt.% water, P3: Tg = 118oC , 6.2wt.% water, P4: Tg =128oC , 1.2wt.% water.

PSfrag replacements 200µm

Figure 5.14: Magnification of the SEM micrograph of P3, Tg = 118oC , 6.2wt.% water expanded at150oC .

5.4 Conclusions

We successfully used the industrially surfactant SURF to prepare a new generation of WEPSbeads. This surfactant resulted in a larger water droplet size in the expandable beads thanachieved by using the earlier applied surfactant AOT, but the suspension stability and repro-


ducibility increased.An optimum in the water content and droplet dispersion in the final compact beads has beenobserved by increasing the NaCl concentration in the emulsified blowing agent. To investi-gate the influence of melt strength of the polymer matrix on the expansion behavior, gradeswith different molar mass were prepared. We were able to increase the molar mass by de-creasing the radical initiator concentration and the addition of a chain transfer agent (DMP)resulted in a decreased molar mass. Besides changing the molar mass to influence the meltstrength, dissolution of a melt strength ’thickener’, viz. end-capped PPE, was applied. Wewere able to prepare, for the first time, water expandable PS/PPE blends successfully.All prepared samples were expanded using our single bead expander. Our single bead ex-pander appeared to be a fast, reproducible and accurate tool for the online determination ofthe relative volume as function of time (expansion characteristic). From the recorded expan-sions, it was shown clearly that the expansion of WEPS exists of three pronounced stages:(I) induction, (II) processing window, including expansion up to a maximum and initiation ofcollapse and (III) collapse. Whereas for EPS, Stage III was not observed. For WEPS, it wasobserved that in Stage I a substantial loss of blowing agent occurred in contrast with EPS.The overall loss of blowing agent results in an early collapse of WEPS compared to EPS.From our expansion results we can conclude that a higher concentration of water does notresult in a higher maximum expansion, whereas the droplet size and dispersion is found to bemore dominant.For standard WEPS the processing window, referred to as Stage II, is very narrow comparedto EPS. We were able to increase the processing window by increasing the melt strength ofthe matrix via increasing the molar mass of the PS matrix and by adding a melt strength’thickener’.

80 Chapter 5

Chapter 6

Modeling versus experiments

6.1 IntroductionAs stated in Chapter 1, the main objective of this thesis is to gain knowledge concerningthe foaming of polymers in general and more specific of WEPS. To achieve this goal, weperformed both a computational and an experimental analysis. In this chapter, the results ofboth analyses are discussed and compared.

6.1.1 Computational analysisThe single bubble model, as presented in Chapter 2, is the basis of the asymptotical analy-sis and numerical models. The asymptotical analysis is performed using the proposed initialframework combined with the constitutive equations as described by the Newtonian, Hookeanand Maxwell model (see Chapter 3) resulting in the time scales for bubble growth in a propermathematical way. The limitations of this analysis concern the incorporation of more com-plex rheological behaviors, heat and mass transfer.For this reason, a numerical analysis was developed in such a way that a more general frame-work was created with the possibility to take into account more complex rheological be-haviors and mass transfer (see Chapter 4). First of all, the numerical analysis was used tovalidate the asymptotical analysis. The numerical and asymptotical time scales were in goodagreement. Hereafter, the influence of several parameters on the growth of the single bubble,the diffusion of the blowing agent and more complex rheological behaviors (e.g. multimodeMaxwell and extended Pom-Pom) were investigated.

6.1.2 Experimental analysisIn Chapter 5, the synthesis of different water expandable polystyrene (WEPS) grades andderivatives was described. These WEPS beads were expanded using our single bead ex-pander. This expander enabled us to accurately record the relative volume of the bead asfunction of time. As discussed in Chapter 5, the expansion of commercial EPS versus WEPSis different. Several reasons are described, one of the differences is the diffusion of waterversus the diffusion of pentane above the glass transition temperature Tg of the PS bead.As observed in our labscale expander, the expansion of WEPS possesses three pronounced

82 Chapter 6

stages: (I) induction, (II) processing window, including expansion up to a maximum andinitiation of collapse and (III) collapse.In the case of EPS, Stage III is not observed for the applied time scale (tmax ≈ 200s). ForWEPS, the three stages in the expansion process can be influenced by the initial amount ofblowing agent and the melt strength of the polymer matrix.

6.1.3 Comparison of model and experimentNaturally, there will be differences between a mathematical model and reality due to inherentuncertainties in the model. Some types of uncertainties that can be encountered in mathema-tical modeling of a real life phenomenon are:

1. Assumptions adopted during the modeling process.

2. Proper values for the parameters that are part of the mathematical model.

3. The mathematical model cannot always be solved analytically or asymptotically in or-der to evaluate the numerical outcome. In this case the solution has to be approximatednumerically and this might introduce numerical errors, e.g. discretization of continuoustime and space systems, computer round-off, etc.

For above mentioned reasons, we not only performed an asymptotical analysis to validate ournumerical analysis (and vice versa), but we performed also several experiments to compareour computational outcome with experimental results.One of the main assumptions in the model is that we consider a single bubble in an infinitemass of polymer fluid, which is related to a single cell present in our beads. We have to realizethat our expandable polymer systems consist of several bubbles (the domains containing wa-ter as the blowing agent) in close proximity, influencing each other during expansion. For thisreason, we have to underline that the conclusions from our modeling have to be consideredin the proper context.Moreover, we have to emphasize that the numerically used relative radius (R/R0) is the radiusof a single cell surrounded by an infinite envelope of polymer fluid. Experimentally, we usedthe relative volume of the bead (V/V0). There is no direct relation between V/V0 and R/R0and therefore, we only perform a qualitative comparison, clarifying the influence of severalparameters on the experimentally expansion behaviors.

6.2 Computations versus experimentsIn this section a qualitative comparison is made between experimental results and compu-tational outcome for EPS, WEPS with different water content, WEPS with different molarmass and the influence of diffusion respectively.

6.2.1 EPSThe computational outcome concerning the Newtonian model (not including diffusion) isgiven by a continuous growth up to a maximum (which is referred to as Stage II). The max-imum expansion for this case only depends on the amount of blowing agent present in thebubble. This expansion behavior is comparable to the experimentally determined growth

Modeling versus experiments 83

for commercial EPS in our labscale expander. Experimentally, we can observe an inductionperiod (Stage I) followed by a continuous growth (Stage II). In case of the computationaloutcome, we use β(t∗) as the fraction of the blowing agent contributing to the gas pressureinside the bubble. We proposed the following function for β(t∗) (see also Eq. 4.33)

t∗ < aπ : β(t∗) = 12 cos

(t∗a − π

)+ 1

2 ,

t∗ ≥ aπ : β(t∗) = 1,(6.1)

where a determines the rate of increase of the amount of blowing agent contributing to thegas pressure inside the bubble. This can be related to a factor slowing down the expansion(Stage II). Both the computationally and experimentally determined growth as function oftime are given in Figure 6.1.

0 2 4 6 8 100

2

4

6

8

10

12

14

II

PSfrag replacements

t[s]

R/

R0

0 50 100 150 2000

2

4

6

8

10

12

I II

PSfrag replacements

t[s]

V/

V 0

Figure 6.1: Left: The computed relative radius R/R0(t) for: Newtonian model, in the direction of thearrow a = 10, 20, 30, at T = 420K , no diffusion, α = 1. Right: The experimentallydetermined relative volume V/V0(t) for commercial EPS at T = 105oC .

From both plots it is clear that R/R0 and V/V0 show qualitatively a comparable expansionbehavior, but differ in a quantitative way as has been explained before. In case of the numer-ical outcome, only Stage II is observed. Stage I could appear by incorporating heat transferinto the numerical analysis.

6.2.2 WEPS

Diffusion

The diffusion is of great influence on the expansion of WEPS as has been seen from therecordings of a single bead (see Figure 5.4 and 5.3). This is also visualized in the experi-mentally determined relative volume as function of time, where for EPS a continuous growthis visualized (see the right plot of Figure 6.1) even up to t = 200s, whereas for WEPS thegrowth is subsequently followed by a collapse (decrease in volume) of the bead narrowingthe processing window (Stage II). Therefore the maximum expansion not only depends onthe amount of blowing agent present in the bubble, but also on diffusion.

84 Chapter 6

Incorporating diffusion in the numerical analysis shows that much higher diffusion coeffi-cients have to be considered, compared to the achieved values using the proposed fit (Kreve-len van, 1990), as can be found in Appendix A, to find an influence of diffusion viz. collapseof the bubble. In literature (Vrentas and Duda, 1976; Sok, 1994; Frisch et al., 1971), it isdescribed that the free volume theory cannot explain the diffusion involving small molecules,such as water, in amorphous polymers. The minimum void volume necessary for migration ofsmall molecules is less than the average void volume of the system. Another approach wouldbe, to consider the loss of blowing agent in case of WEPS as a combination of effects e.g.diffusion, eruption and percolation. Percolation deals with the effect of varying the numberof interconnections present in the polymer network. With several interconnections present,the blowing agent will be able to escape to the exterior of the bead. The phenomena of perco-lation and eruption result in an open-cell foam, but SEM micrographs of WEPS foam possessa closed cell structure and therefor we conclude that no eruption or percolation occurred.The computational influence of diffusion is visualized in Figure 6.2 for the Newtonian model.

0 2 4 6 8 100

2

4

6

8

10

12

14

II III

PSfrag replacements

t[s]

R/

R0

Figure 6.2: Relative radius R/R0(t) for Newtonian material behavior with varying diffusion coefficientD, in the direction of the arrow respectively 10−9, 10−5, 10−4, 10−3, 10−2, 10−1m2/s,at 420K with kh = 10−7cm3/(cm3 · Pa), α = 1, a = 10.

From above figure we can conclude that D ≈ 10−9m2/s in combination with solubilitykh ≈ 10−7cm3/(cm3 · Pa) (Krevelen van, 1990) results in no visible collapse (no Stage II).Increasing D, Stage III arises. In our further analysis we use D = 10−5m2/s in combinationwith kh = 10−5cm3/(cm3 · Pa), showing Stadium II and Stage III, since kh also influencesthe permeability. The combination of D = 10−5m2/s and kh = 10−5cm3/(cm3 · Pa) resultsin a comparable bubble growth and collapse as found using the above used combinationD = 10−1m2/s and kh = 10−7cm3/(cm3 · Pa).

Water content

The computational outcome concerning the Newtonian model, including diffusion for a rangeof blowing agent concentration (amount of blowing agent is proportional with the fill fractionα), is comparable with the expansion of WEPS beads with different amount of blowing agent.These experiments are discussed in Chapter 5 for the grades R1, R2 and R3. Whereas gradeR3 contains a higher water content than R2, but shows a more irregular droplet dispersion


compared to R1 and R2. Both the computationally and experimentally determined growth asfunction of time are given in Figure 6.3.

0 2 4 6 8 10

1

2

3

4

5

6

7

III II

PSfrag replacements

t[s]

R/

R0

0 5 10 150.5

1.5

2.5

3.5

4.5

I II III

PSfrag replacements

t[s]

V/

V 0Figure 6.3: Left: The computed relative radius R/R0(t) for: Newtonian model, in the direction of

the arrow α = 1, 0.6, 0.2, at T = 420K , D = 10−5m2/s, kh = 10−5cm3/(cm3 · Pa),a = 10. Right: The experimentally determined relative volume V/V0(t) at Tbath = 150oCfor grades R1: �, 2.1wt.% water, R2: ◦, 3.3wt.% water, R3: G, 2.8wt.% water.

Computationally, it has been demonstrated that a higher initial fill fraction α leads to a highermaximum expansion, whereas experimentally, it is shown that for the grades R1 (2.1wt .%water) and R2 (3.3wt .% water) a comparable expandability is found and that R3 (irregu-lar droplet dispersion with 2.8wt .% water) yields a lower expandability. This stresses theconclusion that droplet size and dispersion are dominant factors for the expansion process.Unfortunately, our numerical analysis cannot incorporate this effect and the above describedresult suggests incorporating several bubbles in close proximity in our numerical analysis.The numerically found time scales for expansion (Stage II) are in good agreement with theexperimentally found time scales (4 and 5 seconds). The major difference is that our modeldoes not take heat transfer into account, which is expected to show Stage I.

Melt strength

The computational outcome concerning the Newtonian model, including diffusion for diffe-rent viscosities (η is proportional with the melt strength) of the polymer matrix, is comparableto the expansion of WEPS beads with different melt strength, as has been discussed in Chap-ter 5 for the grades R4 and R5. Both the computationally and experimentally determinedgrowth as function of time are given in Figure 6.4.In Figure 6.4 a remarkable qualitative similarity is visualized. As well for the plot on theright as on the left, an increase in melt strength results in an increase in the width of StageII (experimentally observed approximately 6 seconds), the processing window, and a slowercollapse. The experimentally observed difference in maximum expandability is very small.The droplet size and dispersion are expected to play a dominant role in governing the maxi-mum expandability.Another tool (as described in Chapter 5) to increase the melt strength is to prepare waterexpandable blends of polystyrene with PPE, referred to as WE(PS/PPE). Due to the lack of

86 Chapter 6

0 2 4 6 8 10

1

2

3

4

5

6

7

8III II

PSfrag replacements

t[s]

R/

R0

0 5 10 150.5

1.5

2.5

3.5

4.5 I II III

PSfrag replacements

t[s]

V/

V 0

Figure 6.4: Left: The computed relative radius R/R0(t) for: Newtonian model, in the direction of thearrow η = 5 · 104, 1 · 105, 5 · 105, at T = 420K , D = 10−5m2/s, kh = 10−5cm3/(cm3 ·Pa), a = 10, α = 1. Right: The experimentally determined relative volume V/V0(t) forWEPS-grades R4: �, Mw = 237kg/mol , R5: ◦, Mw = 163kg/mol .

input parameters concerning this blend, a comparison between computational outcome andexperimentally determined expansion characteristics is not possible yet.

6.3 Conclusions and recommendationsWith a relatively simple model, we are able to predict experimentally observed results. Oneof the most striking results is the fact that the diffusion of the blowing agent is determining thetypical expansion behavior of WEPS. Whereas for EPS we obtained two stages of expansion,the induction time and the expansion of the bead, for WEPS an additional third stage occurs,the collapse of the bead. The collapse of the bead is a direct result of the loss of blowingagent, as has been observed in our experiments and computations.In our computational model, increasing the amount of blowing agent results in a higher max-imum expansion, this was not observed experimentally. From this, we can conclude that thedroplet size and dispersion in the compact beads are more dominant.The single bubble is able to qualitatively describe the influence of the melt strength of thepolymer matrix on the expansion. With increasing melt strength, the experiments and thecomputations showed an increase in Stage II and a slowdown in collapse (Stage III).We can conclude from the in this chapter described comparison, that to understand foamingin more detail, it is necessary to extend the used single bubble model and to gain more accu-rate input parameters. An improvement would be to gain more experimental knowledge bypreparing more different grades and determining their expansion characteristics.The used single bubble incorporating mass transfer (diffusion) could be extended by incor-porating heat transfer. This extension is expected to introduce Stage I in our numerical re-sults. Another improvement would be considering a multi-bubble approach, in which theinteractions between the different bubbles are incorporated, resulting in a more quantitativedescription of the expansion. The multi-bubble approach needs a more elaborate numericalapproach based on e.g. a boundary integral method (Bazhlekov, 2003) or a diffuse-interfacemodel (Verschueren, 1999). As an ultimate goal, we see a three dimensional representation


of an expanding polymer sphere containing multiple bubbles influencing each other uponexpansion.Concerning the input parameters, it would be very useful to accurately measure the diffusioncoefficients and solubilities of water in polystyrene at temperatures above the glass transitiontemperature of polystyrene. Also data concerning the elongation of polystyrene are necessaryas input for more complex rheological descriptions of the polymer.The last recommendation concerns the experimental part of this thesis. To validate the (fu-ture) numerical analysis in more detail, more experiments are necessary, investigating theinfluence of e.g. temperature, extreme water contents and dispersions, different polymer ma-trices, etc.

88 Chapter 6

Appendix A

Mass diffusion

To determine the decrease in amount of gas present in the bubble, the principle of conserva-tion of mass is applied to the gas inside the bubble. The concentration gradient at the cellboundary can be related to the change in gas amount in the bubble

ddt

(ρg R3

)= 3ρB A DR2 ∂ci(r, t)

∂r|r=R, (A.1)

where ρg is the density of the gas inside the bubble, ρB A is the density of the blowing agentat STP, D is the diffusion coefficient of the gas through the polymer, and ∂ci(r, t)/∂r |r=R isthe concentration gradient of the gas at the bubble interface. The left-hand-side of Eq. A.1 isthe rate of accumulation of mass inside the bubble (which is negative due to loss of blowingagent to the surroundings) and the right-hand-side is the rate of diffusion of gas from theinside of the bubble to the surrounding polymer matrix.Here, we assume that the concentration of the gas at the bubble interface is related to the gaspressure inside the bubble through Henry’s law (Krevelen van, 1990)

cw = ci (R, t) = kh pg(t), (A.2)

where cw = ci(R, t) is the concentration of the dissolved gas in the polymer fluid at thebubble interface and kh is the Henry’s Law constant.The initial amount of blowing agent present in the bubble relates to an initial gas pressure, p0,through Henry’s law. Upon bubble growth, the pressure inside the bubble decreases accordingto the ideal gas law. Consequently, the concentration of the blowing agent present in thepolymer fluid around the bubble decreases accordingly. The presence of the blowing agentin the polymer fluid around the bubble creates a concentration gradient in the fluid aroundthe bubble, invoking the diffusion of the blowing agent from the bubble to the surrounding.The diffusion of the gas through the polymer matrix is governed by the following equation inspherical coordinates

∂ci(r, t)∂ t

+ vr∂ci(r, t)

∂r=

Dr2

∂

∂r

(r2 ∂ci(r, t)

∂r

). (A.3)

To solve Eq. A.1 for the gas pressure inside the bubble, we need to know the concentrationgradient at the interface. Therefore, it is necessary to solve Eq. A.3 to obtain the concentration

90 Appendix A

profile. There are different approaches for solving the diffusion equation, in what follows,the approximate method of solution proposed by Han and Yoo (1981) is used.Multiplying both sides of Eq. A.3 by r 2 and integrating the resulting equation with respect tor between r = R to r = R + δ results in:

∫ R+δ

Rr2 ∂ci(r, t)

∂ tdr +

∫ R+δ

Rr2vr

∂ci(r, t)∂r

dr = D∫ R+δ

Rd(

r2 ∂ci(r, t)∂r

), (A.4)

where δ is the thin concentration boundary layer. After integrating each term of Eq. A.4 usingthe following conditions: r 2vr = R2 R, ci (R + δ, t) = c0, ci(R, t) = cw, ∂ci(r, t)/∂r = 0 atr = R + δ, we obtain

ddt

∫ R+δ

Rr2 (ci(r, t) − c0) dr = −DR2

(∂ci(r, t)

∂r

)

r=R, (A.5)

where c0 is the concentration of the blowing agent in the melt far from the bubble, which isconsidered to remain constant during bubble growth. For now, we assume the temperaturein the bubble as homogeneous and independent of time. Combining Eq. A.1 and A.5, andintegrating with respect to time t yields

(ρg R3 − ρg,0 R3

0

)= 3ρB A

∫ R+δ

R(c0 − ci (r, t)) r2dr, (A.6)

where ρg,0 is the density of the blowing agent in the bubble at t = 0, which is equal to thedensity of the blowing agent at STP (ρ), and R0 is the initial bubble radius. The followingconcentration profile can be used:

c0 − ci (r, t)c0 − cw

= (1 − 5)2 , (A.7)

for R ≤ r ≤ R + δ, in which

5 =r − R

δ. (A.8)

Substituting Eq. A.7 into the right-hand-side of Eq. A.6 leads to

3R3

[1

30

(δ

R

)3

+16

(δ

R

)2

+13

(δ

R

)]=

ρg R3 − ρg,0 R30

ρ(c0 − cw), (A.9)

solving this equation results in

ddt

(ρg R3

)=

6ρB A DR2(c0 − cw)

δ, (A.10)

where δ is the thin concentration boundary layer and c0 the concentration of the blowingagent in the melt far from the bubble, which is considered to remain constant during bubblegrowth. From the approximate method we found

δ = −5R3

+5 · 21/3R2

3C−

C3 · 21/3 , (A.11)

Mass diffusion 91

where

C3 = −270B − 200R3 +√

500R6 + (−270B − 200R3)2, (A.12)

and

B =ρg R3 − ρg,0 R3

0ρ(c0 − cw)

. (A.13)

A.0.1 Henry’s constant and diffusion coefficientAn important property for the gas polymer system needed for the numerical simulation isthe Henry’s law constant. Solubilities of gases in polymer systems depend strongly on thetemperature of the system.In Pogany (1976) we found three values for the solubility and for the diffusion coefficient asfunction of temperature, these values are given in Table A.1

Table A.1: Solubility kh and diffusion coefficient D for the water polystyrene system (Pogany, 1976).

T [K ] kh[cm3/cm3 · Pa] D[cm2/s]295 2.89 · 10−5 1.7 · 10−7

323 1.06 · 10−5 5.6 · 10−7

353 3.98 · 10−5 16.0 · 10−7

We can find a fit for kh as function of temperature by using Krevelen van (1990). Plotting theproposed fit and the values found by Pogany (1976) yields Figure A.1.

300 350 400 450 50010

−12

10−10

10−8

10−6

10−4

10−2

PSfrag replacements

temperature [K ]

solu

bilit

y[c

m3 /

cm3·P

a]

Figure A.1: Solubility kh as function of temperature T for the water polystyrene system (solid line)and the dashed lines are the maximum and minimum according to Krevelen van (1990),the ◦ according to Pogany (1976).

From Figure A.1 we can conclude that the values according to Pogany (1976) and the byKrevelen van (1990) proposed fit match.Another important property for the gas polymer system needed for the numerical simula-tion including diffusion is the diffusion coefficient. In literature experimental values have

92 Appendix A

been found for the diffusion coefficient of the water polystyrene system (Pogany, 1976), seeTable A.1. With these data and the following equation (Krevelen van, 1990)

log D(T ) = log D0 −435K

T· 10−3 ED

Rg, (A.14)

where D0 is a constant for a particular gas and polymer, E D is the activation energy ofdiffusion and Rg is the gas constant, we can find that log D0 = −0.844 and ED/Rg =4.019 · 103.Plotting the diffusion as function of temperature according to Pogany (1976) and to Kreve-len van (1990) yields Figure A.2.

300 350 400 450 50010

−7

10−6

10−5

10−4

10−3

PSfrag replacements

temperature [K ]

diff

usio

nco

effic

ient

[cm

2 /s]

Figure A.2: Diffusion coefficient D as function of temperature T for the water polystyrene system,dashed lines are the maximum and minimum according to Krevelen van (1990), the solidline is according to Pogany (1976).

We realize that the proposed data as function of temperature above the glass transition tempe-rature Tg of polystyrene and above the boiling point Tb of water, both approximately 100oC ,are not valid. In our numerical analysis (Chapter 4) we will use the proposed data only asstarting points, from which we start to investigate the influence of both variables.

Appendix B

Asymptotical analysis consideringthe multimode Maxwell model

For multimode Maxwell, the constitutive equation is written as

λ0,idτ i

dt+ τ i = 2η0,i D, (B.1)

where the viscoelastic contribution of the i th relaxation mode is denoted by τ i . The stressesin the fluid can now be described as

τ =N∑

i=1

τ i , (B.2)

where N denotes the total number of different modes present. Incorporating this model intoour momentum equation results in a similar equation as achieved for the simple Maxwellmodel, but now including a summation as presented in Eq. B.2.Starting with Eq. 3.10 and including the summation leads to

ρReq u(t) = ϕ′(Req)u(t)−

N∑

i=1

4G0,i

Req

[u(t) − u(0)e−t/λi −

1λ0,i

∫ t

0u(t ′)e(t ′−t)/λ0,i dt ′

].

(B.3)

Performing a Laplace transformation (as presented in Chapter 3) results in

ρReq

∫ ∞

0u(t)e−stdt = ϕ′(Req)

∫ ∞

0u(t)e−stdt

−N∑

i=1

4G0,i

Req

∫ ∞

0u(t)e−stdt −

(u(0)λ0,i +

∫∞0 u(t ′)e−st ′dt ′

)

sλ0,i + 1

.

(B.4)

94 Appendix B

Now, u(t ′) and u(t) are transformed to the Laplace domain as follows

u(s) =∫ ∞

0e−st ′u(t ′)dt ′ =

∫ ∞

0e−stu(t)dt . (B.5)

The full equation for u(s) reads

ρReq

(s2u(s) − su(0) − u(0)

)=

ϕ′(Req )u(s) −N∑

i=1

4G0,i

Req

(u(s) − u(0)

λ0,i

sλ0,i + 1−

1sλ0,i + 1

u(s))

,

(B.6)

where the left hand side is obtained by (successive) partial integration. Solving Eq. B.6 foru(s), one finds

u(s) =

ρR2eq u(0) + ρR2

eqsu(0) +N∑

i=1

4G0,iu(0)λ0,i

sλ0,i + 1

ρR2eqs2 − ϕ′(Req)Req +

N∑

i=1

4G0,i

[1 −

1sλ0,i + 1

] . (B.7)

We can use the factor

N∏

i=1

(sλ0,i + 1), (B.8)

to simplify the denominator to a polynomial with order (N + 2).

Bibliography

Agarwal, U. (2002). Simulation of bubble growth and collapse in linear and pom-pom poly-mers. e-Polymers, 14.

Amon, M. and Denson, C. D. (1984). A study of the dynamics of foam growth: analysis ofthe growth of closely spaced spherical bubbles. Polym. Eng. Sci., 24, 1026–1034.

Amon, M. and Denson, C. D. (1986). A study of the dynamics of foam growth: simplifiedanalysis and experimental results for bulk density in structural foam molding. Polym. Eng.Sci., 26, 255–267.

Arefmanesh, A. (1991). Numerical and experimental study of bubble growth in highly viscousfluids. Ph.D. thesis, University of Delaware.

Arefmanesh, A., Advani, S. G., and Mischaelides, E. E. (1990). A numerical study of bubblegrowth during low pressure structural foam molding process. Polym. Eng. Sci., 30, 1330–1337.

Babley, C. J. (1990). Control of voc emissions from polystyrene foam manufacturing, EPA-450/3-90-020. U.S. Office of Air Quality and Standards.

Bazhlekov, I. (2003). Non-singular boundary integral method for deformable drops in viscousflows. Ph.D. thesis, Eindhoven University of Technology.

Benning, C. J. (1969). Plastic foams: the physics and chemistry of product performance andprocess technology. Wiley - Interscience, London.

Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960). Transport phenomena. John Wiley,New York.

Brody, A. L. and Marsh, K. S. (1997). Encyclopedia of Packaging Technology, second edition.Wiley - Interscience, New York.

Cohen, J. (2001). U.S. high GWP gas emissions 19902010: Inventories, projections, andopportunities for reductions, EPA-000-f-97-000. U.S. Office of Air and Radiation.

Crevecoeur, J. J. (1997). Water expandable polystyrene (WEPS). Ph.D. thesis, EindhovenUniversity of Technology.

Crevecoeur, J. J., Neijman, E., Nelissen, L., and Zijderveld, J. (1996). Process for the prepa-ration of polymer particles. Eur. Pat. Appl. 96201904.8.

96 Bibliography

Crevecoeur, J. J., Nelissen, L., and Lemstra, P. J. (1999a). Water expandable polystyrene(weps) part 1. strategy and procedures. Polymer, 40, 3685–3689.

Crevecoeur, J. J., Nelissen, L., and Lemstra, P. J. (1999b). Water expandable polystyrene(weps) part 2. in-situ synthesis of (block)copolymer surfactants. Polymer, 40, 3691–3696.

Eckstein, A., Suhm, J., Friedrich, C., Maier, R. D., Sassmannshausen, J., Bochman, M.,and Mulhaupt, R. (1998). Determination of plateau moduli and entanglement molecularweight of isotactic, syndiotactic, and atactic polypropylenes synthesized with metallocenecatalysts. Macromolecules, 31, 1335–1340.

Fen-Chong, T., Herve, E., Dubault, A., and Halary, J. L. (1999). Viscoelastic characteristicsof pentane-swollen polystyrene beads. J. Appl. Polym. Sci., 73, 2463–2472.

Ferry, J. D. (1980). Viscoelastic properties of polymers. John Wiley and Sons, New York.

Forman, C. (2001). RP-120X polymeric foams - updated edition. Business CommunicationCo.

Frisch, H. L., Klempner, D., and Kwei, T. K. (1971). Modified free-volume theory of pene-trant diffusion in polymers. Macromolecules, 4, 237–238.

Frisch, K. C. (1972). in Plastic Foams, Part I. Marcel Dekker, Inc., New York.

Gietl, A. and Honl, H. (2001). Kunststoffe Plast Europe vol. 91. Hanser, Munchen.

Gluck, G., Hahn, K., and Gellert, R. (1999a). Verfahren zur herstellung wasserexpandierbarerstyrolpolymerisate. Deutsches Patent 198 12 856 A 1.

Gluck, G., Hahn, K., and Gellert, R. (1999b). Verfahren zur herstellung wasserexpandierbarerstyrolpolymerisate. Deutsches Patent 198 12 857 A 1.

Gluck, G., Hahn, K., and Dodel, P. (1999c). Verfahren zur herstellung wasserexpandierbarerstyrolpolymerisate. Deutsches Patent 198 13 108 A 1.

Gluck, G., Hahn, K., and Gellert, R. (1999d). Wassergeschaumte polystyrolpartikel.Deutsches Patent 198 12 854 A 1.

Grulke, E. A. (1990). Encyclopedia of polymer science and engineering. John Wiley andSons, New York.

Han, C. D. and Yoo, H. J. (1981). Studies on structural foam processing iv. bubble growthduring mold filling. Polym. Eng. Sci., 21, 518–533.

Joshi, K., Lee, J. G., Shafi, M. A., and Flumerfelt, R. W. (1998). Prediction of cellularstructure in free expansion of viscoelastic media. J. Appl. Polym. Sci., 67, 1353–1368.

Keppler, H. G., Stahnecker, E., and Moeller, R. (1977). Manufacture of expandable styrenepolymers. United States Patent 4,036,794.

Klempner, D. and Frisch, K. C. (1991). Handbook of Polymeric Foams and Foam Technology.Hanser Publishers, New York.

Bibliography 97

Krevelen van, D. W. v. (1990). Properties of polymers. Elsevier Science Publishers, Amster-dam.

McLeish, T. C. B. and Larson, R. G. (1998). Molecular constitutive equations for a class ofbranched polymers: The pom-pom polymer. J. Rheol., 42(1), 81–110.

Nelissen, L., Nies, E., and Lemstra, P. J. (1990). Influence of pressure on the phase be-haviour of polymer blends: polystyrene/poly(2,6-dimethyl-1,4-phenylene ether). Polymercommunications, 31, 122–123.

Nelissen, L. N. I. H. and Zijderveld, J. M. (1990). Process for preparation of modifiedpolyphenylene ether or related polymers and the use thereof in modified high temperaturerigid polymer of vinyl substituted aromatics. European Patent 0,350,137.

NOVA Chemicals Breda (2002). Private communication.

Pogany, G. A. (1976). Anomalous diffusion of water in glassy polymers. Polymer, 17, 690–694.

Pohleman and Echte (1981). Fifty years of polystyrene. Polym. Sci. Overview, ACS Sympo-sium Series, 175, 3685–3689.

Ramesh, N. S., Rasmussen, D. H., and Campbell, G. A. (1991). Numerical and experimentalstudies of bubble growth during the microcellular foaming process. Polym. Eng. Sci., 31,1657–1663.

Sok, R. M. (1994). Permeation of small molecules across a polymer membrane: a computersimulation study. Ph.D. thesis, University of Groningen.

Street, J. R., Fricke, A. L., and Reiss, L. P. (1971). Dynamics of phase growth in viscous,non-newtonian liquids. Ind. Eng. Chem. Fundam., 10, 54–64.

Swartjes, F. H. M. (2001). Stress induced crystallization in elongational flow. Ph.D. thesis,Eindhoven University of Technology.

Verbeeten, W. M. H., Peters, G. W. M., and Baaijens, F. P. T. (2001). Differential constitutiveequations for polymer melts: The extended pom-pom model. J. Rheol., 45(4), 823–844.

Verschueren, M. (1999). A Diffuse-Interface Model for Structure Development in Flow. Ph.D.thesis, Eindhoven University of Technology.

Vrentas, J. S. and Duda, J. L. (1976). Diffusion of small molecules in amorphous polymers.Macromolecules, 9, 785–790.

Wittenberg, D., Hahn, K., Guhr, U., Hintz, H., O’Callaghan, M., and Riethues, M. (1992).Expandable styrene polymers. United States Patent 5,096,931.

Wu, S. (1970). Surface and interfacial tensions of polymer melts (polystyrene). J. Phys.Chem., 74, 632–638.

98 Bibliography

Technology assessment

Polystyrene foam (’styrofoam’) is a well-known product in society. It is produced by heatingpolystyrene (PS) beads, which are prepared in a reactor via a suspension-polymerization pro-cess, containing pentane-isomers. Upon heating these beads with steam, expansion occurssince pentane with a low boiling point (approximately 30oC) builds up an internal pressureand acts as a so-called blowing agent to expand the PS matrix. Due to partial miscibility ofpentane and PS, plasticizing of the PS matrix occurs, favorable for expansion, and the glasstransition temperature Tg of PS is lowered significantly. During expansion, pentane evapo-rates and the Tg increases to its normal value of approximately 100oC . Pentane, however,is a VOC (Volatile Organic Compound) and contributes to the photochemical smog forma-tion in the lower atmosphere. In the year 2000, the worldwide emission was approximately140ktonnes which makes the current expansion technology not sustainable.In this thesis, the use of water as a blowing agent is studied. Water is environmentally benignand, in principle, a more effective blowing agent than pentane in view of its lower molarmass, 18 versus 72g/mol, viz. in principle 4 times more effective.Replacing pentane with water is non-trivial at all in view of the simple fact that water isimmiscible with PS. In order to reach the goal of replacing pentane with water, one has tomeet a number of criteria, such as:

1. To obtain a discrete dispersion of water droplets in the polystyrene beads.

2. Steam, the commercially used heat source can not be used since the temperature ofsteam and the glass transition temperature Tg of PS are virtually identical, viz. 100oC .Consequently, expansion should occur above Tg.

3. In contrast with pentane, where the Tg increases during expansion (hence stabilizes thematrix), expansion with water as the blowing agent has to be performed above the Tgand the expanded beads are not intrinsically stabilized.

Taking these criteria into account with another unknown parameter, the loss of water duringexpansion reducing the internal pressure and resulting in a possible collapse of the expandedbeads, we used modeling in combination with experiments to explore the key parameters forwater expandable PS (WEPS).Polystyrene is usually prepared via a radical polymerization of styrene. For expandable po-lystyrene (EPS), the blowing agent pentane is added after a suitable conversion. In the caseof WEPS, prior to the suspension polymerization, the water droplets are dispersed in a pre-polymerized styrene/PS mixture possessing a high viscosity in order to fixate the emulsifiedwater droplets. In the thesis, the original recipe (Crevecoeur et al., 1999a) is modified withrespect to the type of surfactant and the suspension stabilizer to increase reproducibility and

100 Technology assessment

stability of the system. The used recipe is promising for industrial applicability since it isproved to be applicable on semi industrial scale (pilot plant vessel 200l). Another advantageis that the EPS reactor setup can be used with few modifications.Expansion experiments in our developed labscale expander using hot oil as heating mediumshowed that the prepared WEPS beads are expandable. Three pronounced stages have beenobserved: (I) induction, (II) processing window, growth of the bead and initiation of collapse,and (III) collapse.Rather unexpectedly, the expandability of WEPS is low compared with EPS at comparablevolume fractions of blowing agent, viz. water versus pentane. The reason is a rather high rateof diffusion of water through polystyrene at temperatures above the glass transition tempera-ture. This premature loss of water results in collapse of the bead (Stage III).In actual practice this diffusion might be suppressed by the use of superheated steam asheating medium. Another possibility to increase the processing window is by increasing themelt strength of the polymer matrix. The melt strength of the polystyrene matrix can beincreased by increasing the molar mass. Upon expansion, this will result in a slower increasein volume (Stage II) but more important in a slower collapse of the beads (Stage III).To increase the melt strength in another way, poly(2,6-dimethyl-1,4-phenylene ether) (PPE)was dissolved in the monomer styrene prior to the pre-polymerization. This results in aPS/PPE matrix, or WE(PS/PPE). In this case the PPE (intrinsic viscosity 46cm3/g) acts asa melt strength ’thickener’. Expansion of the WE(PS/PPE) beads in our labscale expandershowed an increase in the processing window.Summarizing, replacing pentane with water as the blowing agent in the production of poly-styrene foam is feasible, albeit with major modifications in the foaming process. The loss ofwater during expansion, premature collapse of the expanded beads and the size of the dis-persed water droplets are all important factors to reckon with. Last but not least, the use ofPPE of suitable molar mass to increase the melt strength might be of value for the presentcommercial PS/PPE foams, Dythermr, where also pentane is used as the blowing agent.

Samenvatting

Het alom bekende ’piepschuim’ (polystyreenschuim) wordt gevormd uit expandeerbare po-lystyreen (EPS) korrels (’beads’, diameter 0.7 − 1.5mm). Dit halffabrikaat bevat pentaan-isomeren als fysisch blaasmiddel. Ten gevolge van het verwarmen van deze korrels, metbijvoorbeeld stoom, boven de glasovergangstemperatuur (Tg) en boven het kookpunt van hetblaasmiddel (Tbp), treedt expansie op. Tijdens het expansieproces komt pentaan in het milieuterecht. Koolwaterstoffen (in de literatuur aangeduid met VOC = Volatile Organic Com-pound), zoals pentaan, dragen bij aan de foto-chemische smogvorming in de lage atmosfeer.In het midden van de negentiger jaren zijn de toepassingen van deze koolwaterstoffen onderde aandacht gebracht. Sinds die tijd dient het gebruik en ook de emissie sterk teruggedrongente worden (KWS2000). In ons laboratorium is in die tijd een nieuw concept ontwikkeld voorexpandeerbaar polystyreen, waarbij gebruik wordt gemaakt van water als blaasmiddel. Ditproduct wordt water expandeerbaar polystyreen (WEPS) genoemd.Bij het vergelijken van EPS met WEPS valt direct op dat niet alleen sprake is van praktische,maar ook van fundamentele verschillen. Een belangrijk verschil is dat water niet oplosbaaris in polystyreen, terwijl pentaan wel oplost in polystyreen. Hierdoor dient water discreetgedispergeerd te worden in de polystyreen korrels. Een ander gevolg is dat pentaan een Tgverlagend effect heeft op polystyreen, terwijl water dit niet heeft. Door diffusie vermindertde pentaan hoeveelheid tijdens de expansie en verhoogt zo weer de Tg. Een ander verschil isdat in het geval van EPS initieel een verschil (∼ 40K ) bestaat tussen het kookpunt van hetblaasmiddel en de Tg van de bead, terwijl in het geval van WEPS nauwelijks sprake is vanzo’n verschil. Ten gevolge van dit laatste dient een gewijzigd expansie-protocol te wordentoegepast. Om de genoemde verschillen op een betere manier te begrijpen, te controleren enom bestaande processen te verbeteren en nieuwe processen te ontwikkelen, is het schuimenvan polymeren in het algemeen en van WEPS in het bijzonder gemodelleerd. Het ’single bub-ble model’ is in dit proefschrift gebruikt voor de gepresenteerde asymptotische en numeriekeanalyse. Dit model maakt gebruik van een cel in een oneindige hoeveelheid materiaal, bi-jvoorbeeld een polymeer.De asymptotische analyse start met een niet-lineaire integro-differentiaalvergelijking. Dezevergelijking volgt direct uit de bewegingsvergelijkinggecombineerd met de continuıteitsverge-lijking en het gebruikte deformatiegedrag. De analyse maakt gebruik van visceus, elastischen visco-elastisch deformatiegedrag. Voor het visceuze en het elastische gedrag wordt re-spectievelijk gebruik gemaakt van de wet van Hooke en Newton. Het visco-elastische de-formatiegedrag wordt beschreven met behulp van een Maxwell-element. Met behulp van deasymptotische analyse worden de specifieke tijdschalen voor celgroei analytisch bepaald, ditvereist een uitgebreide mathematische aanpak. De aanpak omvat een evenwichtsbenadering,een Laplace transformatie en een eigenfrequentie probleem. Dit resulteert in een kubische

102 Samenvatting

vergelijking die, met behulp van een kwadratische analyse, wordt getransformeerd naar eenset analytisch oplosbare vergelijkingen. De oplossingen van deze vergelijkingen bevatten detijdschalen voor celgroei.De numerieke analyse wordt uitgevoerd om de asymptotische analyse te valideren (en viceversa) en tevens om een numeriek raamwerk te creeren waarin meer complex deformatiege-drag en massatransport (diffusie) kunnen worden bestudeerd. De numerieke resultaten gevenduidelijk aan dat de diffusie van het blaasmiddel een belangrijke invloed heeft op het expansie-gedrag, terwijl andere variabelen, zoals viscositeit, modulus, externe druk en blaasmiddelcon-centratie vooral invloed hebben op de tijdschalen voor celgroei en maximale expansie.Om onze asymptotische en numerieke analyse te onderbouwen, zijn de resultaten van dezeanalyses vergeleken met experimentele expansies van water expandeerbare systemen. Hetbestaande WEPS-procede, een pre-polymerisatie (dispergeren van water in styreen) gevolgddoor een radicaalpolymerisatie in een suspensie-proces, is verbeterd met betrekking tot sus-pensiestabiliteit en reproduceerbaarheid. Met dit recept werd een serie WEPS-korrels bereid,varierende in hoeveelheid water en molgewicht. Dit molgewicht is gerelateerd aan de smelt-sterkte van de polymeer matrix. Een andere manier om deze smeltsterkte te beınvloeden is hetoplossen van poly(2, 6-dimethyl-1, 4-fenyleen ether) (PPE) in het monomeer styreen voor depre-polymerisatie. Dit resulteert in de water expandeerbare, mengbare blend van PS en PPE,afgekort met WE(PS/PPE). Dit product is voor de eerste keer succesvol gesynthetiseerd.Alle water expandeerbare types werden geexpandeerd met behulp van de ontwikkelde labo-ratorium opschuimopstelling (V ≈ 10ml). Het schuimproces werd online geregistreerd metbehulp van een digitale camera en resulteerde uiteindelijk in expansiekarakteristieken. Deverkregen curves zijn opgebouwd uit drie stadia: (I) initiatie, (II) schuimproces (expansie eninitiatie van volume-afname) en (III) ’collapse’, afname in volume.Tenslotte werd de numeriek bepaalde celgroei kwalitatief vergeleken met de experimenteelverkregen expansiekarakteristieken. Als diffusie wordt meegenomen in de numerieke be-rekeningen wordt Stadium III verkregen. Numeriek werd gevonden, dat het verhogen vande blaasmiddelconcentratie de maximale expansie doet toenemen, echter dit wordt experi-menteel niet waargenomen. Dit wordt veroorzaakt door de dominante invloed van de water-druppelgrootte en -verdeling, deze varianten zijn niet opgenomen in het model. Het verhogenvan de smeltsterkte resulteert in een verbreed Stadium II en een lagere collapse-snelheid; ditwerd zowel experimenteel als numeriek waargenomen.Concluderend, ondanks het feit dat de modellering is beperkt tot het ’single bubble model’heeft dit, door gebruik te maken van de vergelijking tussen experimenten en modellering,waardevolle resultaten opgeleverd. Het verhogen van de smeltsterkte door het toevoegenvan PPE, van het juiste molgewicht, is van belang voor het commerciele PS/PPE schuim,Dythermr, aangezien dit product ook gebruik maakt van het blaasmiddel pentaan.

Dankwoord

Dit proefschrift is het resultaat van bijna vijf jaar onderzoek binnen de groep Polymeer Tech-nologie aan de Technische Universiteit Eindhoven. Op deze plaats wil ik iedereen bedankendie, direct dan wel indirect, een bijdrage hebben geleverd aan de totstandkoming. Een aantalwil ik in het bijzonder noemen.

Allereerst wil ik natuurlijk Piet Lemstra en Laurent Nelissen bedanken omdat ze mij dekans hebben geboden mijn promotieonderzoek uit te voeren. Piet, bedankt dat je mij demogelijkheid hebt gegeven om naast onderzoekskwaliteiten ook organisatorische kwaliteitente ontwikkelen in zowel de organisatie van de India-reis (2000), als de organisatie van het Eu-roPolymer Congress (2001). Mede dankzij deze ervaringen is mijn promotietijd een geweldigeen afwisselende periode geworden. Laurent, bedankt voor de prettige samenwerking, je altijdkritische blik en je onvermoeibare correcties op meer dan alleen mijn proefschrift. De velegezamenlijke bezoeken aan onder meer NOVA waren een welkome afwisseling.

Guy, jij bent diegene geweest die gezorgd heeft voor de ommezwaai in mijn promotie-onderzoek. Mede hierdoor ziet een groot deel van mijn proefschrift eruit zoals het hier ligt.Ons veelvuldige overleg is voor mij een extra stimulans geweest. Guy bedankt! En uiteraardwil ik Dieneke, Frank, Victor, Joren en Seger bedanken voor de gastvrijheid waar ik velemalen gebruik van heb mogen maken.

Patrick, bedankt voor de prettige samenwerking. Zonder jouw enthousiasme en de vele nut-tige tips, met name op het numerieke vlak, was dit werk niet tot stand gekomen.

NOVA Chemicals wil ik bedanken voor het financieren van het onderzoek en met name deresearch groep onder leiding van Michel Berghmans voor hun ondersteuning en begeleidingvan diverse experimenten.

Met veel plezier heb ik de afgelopen jaren gewerkt in de groep SKT. Ik bedank dan ookiedereen die heeft bijgedragen aan de prettige werksfeer. Mijn kamergenoten (in volgordevan vertrek en opkomst), Wilfred, Frank, Sachin en Michael bedank ik voor de gezelligheidbinnen STO 0.43, onze goedemorgen-koffie-corner. Deze gezelligheid ging vaak samen metveel (redelijk sterke) koffie in combinatie met vele collega’s, onder meer Bert, Edgar, Hans,Ilse, Jules, Martijn en Michael. De gevoerde discussies, waarbij ik Peter speciaal wil noemen,hebben vaak geleid tot uiterst interessante oplossingen voor de meest uiteenlopende zaken.Tevens wil ik Pauline bedanken voor het in een razend tempo produceren van de benodigdeSEM foto’s.

104 Dankwoord

En als laatste natuurlijk heel veel dank aan mijn familie en vrienden. Waarbij ik graag eenaantal mensen speciaal wil bedanken: Gerard, Anne, Antoine, Sindy, Opa en Oma voor jullieinteresse en steun en Renske voor jouw altijd aanwezige organisatietalent, maar bovenal jesteun en vertrouwen.

Emile

Curriculum Vitae

The author of this thesis was born in Hengelo (Overijssel), The Netherlands, on January 21,1975. After finishing secondary school in 1993 (VWO, Scholengemeenschap St. Ursula,Horn), he studied Chemical Engineering at the Eindhoven University of Technology. Heobtained his degree in August 1998 on the project ’The Expansion of Water Expandable Po-lystyrene (WEPS)’ under supervision of dr. L. Nelissen and prof.dr. P.J. Lemstra.

In the same year, he started his PhD-study in the department of Polymer Technology headedby prof.dr. P.J. Lemstra. The results of this research are described in this thesis. Duringhis PhD-study, the author completed all modules of the course ’Registered Polymer Tech-nologist’ (RPK, ’Register Polymeerkundige’) organized by the ’National Graduate School ofPolymer Science and Technology’ (PTN, ’Polymeertechnologie Nederland’).

From May 2003, the author will join TNO Industrial Technology in Eindhoven, The Nether-lands.

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